Factorization Calculus and Geometric Probability

P \ In this way the space G attains the product structure G = S1xR,

(2.1.1)

where S t is the unit circle. Because of the correspondence g -> the right halfplane bounded by g the space (2.1.1) can also be considered as the space of closed halfplanes. In this sense the topology of G complies with the space of closed sets on the plane; see §1.5.

2.2 The space G of (non-directed) lines in U2

21

p axis

The group T2 of parallel shifts of IR2 induces a group of transformations of G (which we also call T2). A t e T2 acts on a y e G in such a way that cp remains unchanged (sends a generator into itself). The generators of the cylinder represent the sets of parallel lines. The distances between the parallel lines under any t e T2 remain unchanged. This means that T2 is a group of Cavalieri transformations of S x x IR. Therefore, product measures with Lebesgue factor on R are T2-invariant.

2.2 The space G of (non-directed) lines in U2 This space may be obtained from G by erasing the arrow on each g e G (under which operation every two distinct directed lines, coincident but with opposite arrows, map into a non-directed line). We obtain a topology which complies with the space of closed sets in U2. We can also consider G as a fibered space. Here the fibers are sets of parallel lines, and thus a fixed fiber model is U. A fiber is represented by the line which passes through the origin O eU2 and is orthogonal to the lines of the fiber. We label the lines through O by the points cp e E x (see §1.5), and thus call them \ where & = g n (cp-line) A sequence gn = (n) converges to g = (cp9 @>) if and only if (1) cp = lim n) in G (because cpn can violate (1)). This hints that the topology of G is different from that of U2: one can show that G is homeomorphic to a Mobius band (see also §5.3). Our fibering has the form TT: G

->E 1 ?

n(g) = cp.

22

2 Measures invariant with respect to translations

In this way translations of R2 act on fibered G in Cavalieri fashion. Therefore every composition (in the terminology of §1.5) of Lebesgue measure Lx on fibers is invariant with respect to T 2 .

2.3 The space E of oriented planes in [R3 An oriented plane in R3 is a plane for which one of the two possible normal directions is called positive. Let e be an oriented plane. The closed halfspace bounded by e, in which the positive normal of e lies, will be denoted by e+ (see fig. 2.3.1). Owing to the natural correspondence we can consider E also as the space of halfspaces. We introduce coordinates on E of the form (co, p), where — oo < p < oo and co is a spatial direction in R3, i.e. co belongs to the unit sphere S 2 . For a given plane e e E, p is the signed distance from the origin O in IR3 to e (p > 0 if O e e+; p < 0 if 0 6 e+; and p = 0 if O e e) and co is the positive normal direction. The space of pairs (co, p) is the product S 2 x R. The topological structure on E determined by the product E = S2 x R

(2.3.1)

complies with the topology induced on the set of halfspaces by closed sets in R3 (see §1.4). Since the parallel shifts of R3 preserve the orientations of the planes, as well as the distances between pairs of parallel planes, we conclude that the group T3 of parallel shifts of R3 is a group of Cavalieri transformations on the product (2.3.1). Therefore product measures with Lebesgue factors on R are T3-invariant.

Figure 2.3.1

2.5 The space F of directed lines in U3

23

2.4 The space E of planes in U3 This space may be obtained from E by mapping every two distinct halfspaces with a common boundary into their common boundary. This yields a topology which complies with the space of closed sets in U3. We can also consider E as a fibered space. Now the fibers are sets of parallel planes and thus a fixed fiber model is IR. A fiber is represented by the line which passes through the origin 0 in U3 and is orthogonal to the planes of the fiber. We label the lines through O by points co e E 2 (the elliptical plane, see §1.5), and thus we call them co-lines. The elements of E are now represented as e = (co, &)

where 3P = e n (co-line).

The topology on E is determined by the following convergence rule. A sequence en = (con, &„) converges to e = (co, 0*) if and only if (1) co = lim con in the topology of E 2 ; (2) & = lim 0>n in the topology of IR3. Note that if & # 0, then (2) implies (1); yet lim &n = 0 in U3 does not guarantee convergence of (con, gPn) in E (con can violate (1)). We note that with this topology E is homeomorphic to elliptical (projective) 3-space with one point deleted. Our fibering has the form n : E -> E 2 ,

n(e) = co.

3

Translations of IR act on this fibered E in Cavalieri fashion. Therefore compositions of L1 planted on fibers are all invariant with respect to T 3 .

2.5 The space T of directed lines in IR3 A directed line in IR3 is a line with an arrow on it. In order to describe a y e F we first determine the direction Q of y, Q e S 2 , and then the point & where y hits the plane tangent to § 2 at Q (we assume S 2 to be centered at O). This tangent plane is orthogonal to Q. The representation

y = (a ») maps r in a one-to-one way onto the tangent bundle of S 2 described in §1.5 and induces the topology of the latter space on t. Thus we consider f as a fibered space: 7r:f->§2,

n(y) = Q.

We stress that the fibers in F are the sets of parallel and equidirected lines; the fixed fiber model is (R2.

24

2 Measures invariant with respect to translations

Parallel shifts of 1R3 leave S 2 intact and act as parallel shifts of IR2 on each fiber. Therefore T 3 is a group of Cavalieri transformations of the space F. We conclude that composition measures on F which have the twodimensional Lebesgue measure L 2 on fibers are all T3-invariant.

2.6 The space F of (non-directed) lines in IR3 This space can be obtained from T by identifying two distinct directed lines, coincident but with opposite arrows. The resulting topology complies with the space of closed sets in U3. We can also describe F as a fibered space. The fibers now are sets of parallel lines; a fixed fiber model is IR2. A fiber can be represented by a plane through the origin O e U3 which is orthogonal to the lines of the fiber. We label the planes through O by points of the elliptical plane Q e E 2 and thus call them Q-planes. We represent the elements of F as y = (Q, 0>\ where 9 = y n (the Q-plane). Using the topologies on E 2 and U3 we define a topology on F by means of component-wise convergence. The result complies with the topology of closed sets in (R3. Our fibering has the form TT:F->E2,

7i(y) = Q.

3

Translations of IR act on F in Cavalieri fashion. Therefore composition measures with L 2 on fibers are T3-invariant.

2.7 Measure-representing product models I Let Gx be the space of lines on IR2 which hit the Ox axis (lines parallel to the Ox axis are excluded). Each line g e Gx can be described by two parameters, x, \j/: x is the abscissa of the point g n (Ox axis); ^ is the angle of intersection of g with Ox. Thus x e R, \j/ e (0, n) (see fig. 2.7.1). The map g - • (\jj,

x)

induces a product structure on Gx: Gx = (0, n) x Ox axis. On the other hand, in Gx we can use the (cp, p) coordinates described in §2.1. For this purpose it is enough to convert each line from Gx into a directed line

2.7 Measure-representing product models

25

g

Figure 2.7.1 by putting an arrow on it in some way. We place the arrow so as to have V = *, (2.7.1) i.e. each time the arrow points toward the upper halfplane. This, together with p = x sin ij/,

(2.7.2)

describes the transition from (i//, x) to (
(2.7.3)

We can set a one-to-one mapping between G and E x x U. In addition to cp, p coordinates on Gx we introduce, for this purpose, a p coordinate on the set of horizontal lines, i.e. on the fiber corresponding to the endpoints of (0, n). For the horizontal line with the equation y=c we put p = c9 and this completes the construction of our map. It is discontinuous, yet it has the property that the image of every compact set in G can be covered by a compact set in E x x U and vice versa. The proof follows easily from the following observations. Since the distance of a line g from 0 always equals \p\9 the image of the set (compact in G) Cx = {g e G : the distance of g from 0 ^ x} is a product set (compact in E x x IR) C; = Ei x {/?e(-oo, o o ) : | p | ^ x } . The image of any A a Cx belongs to Cx and that of any B a Cx belongs to Cx. Lastly, any compact A a G can be covered by some Cx, and any compact B a E x x IR can be covered by some Cx. Corollary §1.6)

Under our map E x x IR is a measure-representing model of G (see

26

2 Measures invariant with respect to translations

II Let Ez be the space of planes in U3 which hit the Oz axis (planes parallel to the Oz axis are excluded). Each plane e e Ez can be described by the quantities co and z: z is the abscissa of the point e n (the Oz axis); co is the normal direction to e represented by a point on the open hemisphere S 2 /2. Here S 2 /2 denotes the part of the unit sphere centered at O which lies in the halfspace z > 0. The map e - • (co, z)

induces a product structure on Ez: Ez = S 2 /2 x Oz axis. Since we have now defined co to be a point in S 2 , we have thus mapped E2 on a part of E. Therefore, instead of (co, z) we can use in Ez the coordinates (co, p) defined in §2.3. The relation between p and z is p = z|cos v|,

(2.7.4)

where v is the angle between co and the Oz axis. Clearly the above amounts to E « E 2 x U.

(2.7.5)

We can extend the (co, p) coordinates to the set of planes parallel to the Oz axis. Each plane of the latter set is determined by the line trace it leaves on the xOy plane. Therefore the problem is essentially that of introducing (cp9 p) coordinates in the set of these lines assuming that cp e E x describes a pair of centrally-symmetrical points on the boundary of S 2 /2. We have just solved this problem in subsection I above. After applying this solution here we obtain a one-to-one (discontinuous) map of E onto E 2 x U. By an argument similar to that we used in I above we conclude that under our map E 2 x U is a measure-representing model of E (see §1.6). III Let Txy be the space of lines in U3 which hit the xOy plane (lines parallel to the xOy plane are excluded). Each line y e Txy can be described by Q - the point of intersection of y with the xOy plane; Q - the point where y hits S 2 /2. The map

?->(ae) induces a product structure on Fxy:

2.8 Factorization of measures on spaces with slits

27

Txy = S2/2 x xOy plane. Since we have now defined Q to be a point in S 2 , we have thus mapped Yxy on a part off. Therefore in Txy we can also use the coordinates (Q, @>) defined in §2.5. For fixed Q, 3P is the projection of Q on the plane perpendicular to y. The image of the Lebesgue measure dQ will be d ^ = | cos v| d(),

(2.7.6)

where v is the angle between y and the z axis. We have found that r « E 2 x U2.

(2.7.7)

Our map can be extended in a way similar to that used the previous cases. The result will be that E 2 x U2 is a measure-representing model of T. Remark The space F also possesses a measure-representing product model, namely the space § 2 x U2. IV Translations act on the space Gx, Ez and Txy in Cavalieri fashion. For instance, for t e T2 and (ij/9 x) representing a line from Gx we have t{il/,x) = (il/,t1x), where rx is the projection of t on the Ox axis. Therefore, product measures on these spaces with Lebesgue factors on the Ox axis, Oz axis or xOy plane, respectively, are translation-invariant. Because of the one-to-one nature of the maps, the action of translations on our product models of the spaces G, E and T is well defined. Any translation produces rigid shifts of the Euclidean generators of our models, i.e. is again Cavalieri. Therefore the product measures with Lebesgue factors on Un, n = 1, 2, are translation-invariant. These measures correspond to similar composition measures on G, E or T viewed as fibered spaces. We will see below that here no other T-invariant measures exist.

2.8 Factorization of measures on spaces with slits In the previous section we constructed product representations for the slitted versions GX9 Ez and Txy of the spaces G, E and I\ We can directly apply Lebesgue factorization to these products. The results are in table 2.8.1. In the second column the table lists the groups under which the measures are assumed invariant. Note that these groups are subgroups of the groups of translations of the corresponding spaces Un, n = 2, 3. The striking feature of these factorizations is that they imply invariance with respect to all shifts of corresponding Un (see the remarks at the end of the previous section).

28

2 Measures invariant with respect to translations

Table 2.8.1

Space

Invariance assumed

Factor measures

Invariance that follows

Gx = (0, K) x (Ox axis)

Shifts of IR2 parallel to Ox

Lebesgue Lx on the Ox axis, some measure on (0, n)

All shifts of U2 (the group T2)

Ez = S 2 /2 x (Oz axis)

Shifts of U3 parallel to Oz

Lebesgue L x on the Oz axis, some measure on S 2 /2

All shifts of U3 (the group T3)

Txy = § 2 /2 x (xOy plane)

Shifts of U3 parallel to xOy

Lebesgue L 2 on the xOy plane, some measure on S 2 /2

All shifts of U3 (the group T3)

A similar table can be compiled for the spaces G x , Ez and Txy which are obtained from G, E and F by deleting the elements which do not hit the corresponding line or plane. The spaces (0, n) or S 2 /2 will be replaced by slitted versions of S x or S 2 .

2.9 Dispensing with slits Let /i be a T2-invariant measure on G or equivalently on E x x U (see §2.7). Because for any a > 0 H({9 = ((P,P)>\P\


co

the measure ^ cannot charge more than a countable number of (linear) generators of the cylinder E x x U. Therefore we can locate an axis g0 through O with the property fi({lines parallel to g0}) = 0.

(2.9.1)

Without loss of generality we can (and do) take g0 to be the Ox axis. Because of (2.9.1), \i now can be considered as a measure defined on G*. It is invariant with respect to shifts parallel to any axis, and also therefore to g0 = Ox axis. We can apply the first line of table 2.8.1: in if/, x coordinates on Gx, \x necessarily has the factorized form m(d\j/) Ax.

Using (2.7.1) and (2.7.2) we change to
(2.9.2)

Because \i was originally defined as a (locally-finite) measure on E x x (R we necessarily have

f

Jo

(sin cp) 1m(d(p) < oo.

(2.9.3)

2.10 Roses of directions and roses of hits

29

Table 2.9.1

Space

Measure representing product model

Group

G

T2

Necessary product representation of invariant measures m(d(p) dp, where dp is Lebesgue Lx on U; m is a totallyfinite measure on Ei

E9 x

m(dco) dp, where dp is Lebesgue Ll on U: m is a totally-finite measure on E 2

x

m(dQ) d^, where d0> is Lebesgue L 2 on (R2; m is a totallyfinite measure on E 2

E,

In view of (2.9.1), the expression (2.9.2) provides a complete description of \i as a measure on E x x U. We conclude that \i is necessarily a product measure. A similar argument holds for the spaces E and F. We can compile the following factorization table 2.9.1. A similar table can be compiled for the space G, E and F.

2.10 Roses of directions and roses of hits The measures in the spaces E x and E 2 , which are listed in the last column of table 2.9.1 are called the roses of directions of the corresponding measures in the spaces G, E or I\ Accordingly we call the points of E x and E 2 directions. We denote translation-invariant measures in G, E or T by fi in the following. The above factorizations imply that the values of the measures \i on the sets of the type [<5] = {g eG:g [ / ] = {y

G

hits a line segment d c

r :y hits a flat / c U3}

(a 'flat' is a bounded part of a plane in IR3 (see §2.13)), and [s] = {e e E: e hits a line segment s c R 3 possess a rather special structure. We have =

dpm(d(p)= J[d]

proj(<5)m(d(p), JJS.I

30

2 Measures invariant with respect to translations

where proj(<5) denotes the length of the projection of S on the p axis (the axis perpendicular to the direction cp). If the direction of 5 is a then (n proj(<5) = cos - + cp — a \2 \S\ stands for the length of <5. Thus The function (a) =

r

|sin(o) — a

aeEi

(2.10.1)

is called the rose of hits of the measure //onG (this terminology was introduced in [17]). We could obtain a more usual expression with cos instead of sin if in §2.1 we defined cp to be the direction perpendicular to the line g. A similar result holds for ju([/])- We have f m{&co)&0>=[ proj(/)w(do>), hn JE 2 where proj(/) is now the area of the projection of/ on the plane carrying (whose normal direction is co). Clearly where £ is the spatial direction normal to /, (co, £) is the angle between the two directions, and ||/|| is the area of/ Thus where

-I. r

)E2

^)|m(dco), ^ e E 2 .

(2.10.2)

This function k(£,) is called the rose of hits of the measure fi on T. For T3-invariant measures on E we have (2.10.3) where £ is the spatial direction of the segment s and \s\ is its length. It is not difficult to prove that k(£) in (2.10.3) necessarily possesses the representation (2.10.2). The function k(£) as defined by (2.10.3) is called the rose of hits of a T3-invariant measure on E.

2.11 Density and curvature Let [i be a T2-invariant measure on G and let k(cp) be its rose of hits. We assume now that the rose of directions has a continuous density f{cp\ i.e. dfi = f(cp) dcp dp.

By (1.7.2) and the second part of Minkowski's proposition in §1.7 we conclude

2.12 The roses of T3-invariant measures on E

31

that X(
What is the geometrical meaning of f(cp) in terms of this contour? Using the formulae (3.6.2) and (3.7.4), we find n[(5 2 ]),

(2.11.1)

where the infinitesimal segments <5X and S2 are shown in fig. 1.8.2, [(5J = {the lines that hit the segment <5J. We assume that 5X and S2 are perpendicular to sx and have a common length /, s1 has direction cp, and the length of sx is 1. On the other hand, by a simple application of the combinatorial formula (1.8.3) we find (using the notation of fig. 1.8.2)

/ 2 ) + Hep -

V In conjunction with (2.11.1) this yields

2f(cp) = ^(X(

) = b(q>) + b"(q>\

(2.11.2)

which expresses the curvature radius R of a symmetrical convex contour at the point with normal direction cp in terms of the breadth function. We conclude that R(cp)=f(cp).

(2.11.3)

Conversely, (2.11.2) can be resolved (see [6] for details) to yield b(
Jo

sin(c) — a)|i<(a) da.

This integral is a version of (2.10.1). Therefore b(cp) happens to be the rose of hits for the rose of directions R(cp) dcp. In this way we get an independent partial proof of the assertion in §1.7 about the generation of metrics by measures in G.

2.12 The roses of T3-invariant measures on E Let \i be a T3-invariant measure on E. It generates a pseudometric p in 1R3: where £ is the direction of the segment &x&2, \&i&*2\ *s ^ts length, and

32

2 Measures invariant with respect to translations 0>1\0>2 = {eeE:e

separates ^

from 0>2).

If/x is not concentrated on a bundle of planes parallel to some fixed direction, then p is a metric. Clearly A(£) is the rose of hits corresponding to p. A proof that p as given by (2.12.1) is a pseudometric in U3 follows from the planar result of §1.7, since the image of \i under the map e -> the line ene0

(2.12.2)

3

where e0 is some plane in (R , is a measure in the space of lines on e0. The converse problem: does every T3-invariant continuous, linearly-additive metric p in 1R3 permit a representation (2.12.1) with some measure \i on E? has a negative solution which we outline below. Let usfixa plane e0 a R3 and identify the space of lines on e0 with G. Let fi0 be the measure on G which is the image of \i under (2.12.2). If we assume the existence of the density: d\x = f(co) da> dp

(we use the notation of §2.7), then by the results of §3.6, §3.11 and (3.12.4) we find dfi0 = F(
J<

We stress that the direction q> on e0 determines a direction Q in R3, and in (2.12.4) integration is over the great semicircle orthogonal to the direction Q. The angle if/ is between co e and the direction normal to e0 (the situation is in a sense dual to that shown infig.1.7.2). The condition F(Q)=\

fsin2il/dil/>0

for every eo,\jj

(2.12.5)

suffices to have a continuous, linearly-additive metric in IR3 which on every plane e0 corresponds to the measure on G given by (2.12.3). However, the condition (2.12.5) can be met by smooth functions / which are not everywhere non-negative. For instance, let us take f0 = 1 on the whole hemisphere with the exception of a circle of radius e; in the center of this circle, say, let f0 = — 1 and let | / 0 | ^ 1 everywhere. We leave it to the reader to prove that by appropriate small choice of s any such (smooth) f0 will satisfy (2.12.5). Suppose in (2.12.4) we insert f = f0. Then the corresponding measure p0 will define a metric pe^Px, 0>2)>§'m the plane e0. Clearly

W&2

and this value does not depend on the choice of e0 through ^ \ , ^ 2 - Thus,

2.12 The roses of T3-invariant measures on E

33

will be a metric in (R3. It follows from a uniqueness result discussed in §6.2 that p0 cannot be generated by some measure /z on E by means of (2.12.1). By Minkowski's proposition (which is valid in [R3, see §1.7) the ratio \&u&i\~l Po(^i> ^2) happens to be the breadth function of a convex body in U3. Since this ratio cannot be a rose of hits, we conclude that the body fails to be a zonoid. Examples of non-zonoidal polyhedrons can be easily constructed using the criterion of §5.10. Here we have constructed a non-zonoidal convex body, which is not a polyhedron. Now let us return to (2.12.1) assuming the existence of density f(co). By Minkowski's proposition, k{£) coincides with the breadth function b(£) of some symmetrical convex body D a [R3, i.e.

m = na

(2.12.6)

Now the purpose is to relate the curvatures of dD with / . The restriction of X{^) to directions in e0 is the breadth function of the projection of D on the plane e0 (see the end of §1.7). By (2.11.2) and (2.11.3) the expression / sin2 ij/ d\j/

2R = r(Q) + /l(Q) = 2

(2.12.7) J gives the doubled curvature radius R of the boundary of the aforementioned projection at the point having planar normal direction cp (which corresponds to Q). We stress that the double differentiation above is within the set of directions which belong to the plane e0. Now we fix the direction Q and rotate e0 around Q. The quantities in (2.12.7) depend on O, the angle of rotation. We average (2.12.7) with respect to G>. We have

P A"(Q)dO = -A /l(Q)

(2.12.8)

2

Jo where A2 is the Laplace operator. By an interchange of the order of integration we readily find TT

1

Cn

C dO

Jo

If

/ sin2 ij/ dxjj = 2

J

/ dO.

J

Thus the result of averaging (2.12.7) overr (0, (0, n) n will be ^A2A(Q) + 2(Q) 2

=11

/dO

(2.12.9)

J<«> J<

The expression ^A2>3,(Q) + A(Q) is well known in differential geometry [6]. It equals the sum Rx + R2 of the main curvature radii. Thus we have Rx + R2=

/dO.

J

(2.12.10)

34

2 Measures invariant with respect to translations

As a byproduct the following result concerning the mean values of curvature radii is obtained R(Q>) d O = —

2.13 Spaces of segments and flats I Let Af denote the space of unit length directed segments on IR2 (a directed segment is a linear segment, one of whose endpoints is called a source). A segment 3 e Af can be described as

8 = fo Q\ where QGU2 is the source, and cp e S1 is the direction of 3 (see fig. 2.13.1). Therefore we have AJ = S x x [R2.

(2.13.1)

The space AS also has the following important interpretation: Af = M2, where M2 is the group of all Euclidean motions of the plane. Indeed each motion M e M2 can be identified with 3 = M30, where (50 is the segment with the source at O (the origin in U2) and cp = 0. Lastly A*^GxH

(2.13.2)

since each 3 can be determined by its carrying line g e G and one-dimentional coordinate x of its source on g. It is also possible to define AS to be the space of directed segments of arbitrary fixed length |<5| = const. II The space of all directed segments on IR2 (with varying lengths) can be described as A2 = A* x (0, oo) (2.13.3) (as compared with (2.13.1) one dimension is added to indicate the length / of the segment). We again denote elements of A2 by 3. III The space A? of unit length directed segments in U3 has the following representations:

2.13 Spaces of segments and

flats

35

A* = S 2 x U3

(2.13.4)

Af = f x R,

(2.13.5)

and which are similar to the representations for Af. Warning: we can describe a 3 e Af as 3 = (e9 9) where e e I is a plane through the source of (3 with positive normal vector parallel to 3, and 9 e R2 determines the position of the source of 3 on e. However, Af is not homeomorphic to E x R2. The reader may try to prove that E x U2 is a measure-representing model of Af. Neither can Af be identified with M 3 , the group of Euclidean motions of IR3. In fact, Af is less than M 3 in dimension (see §3.13). As for A3, the space of all directed segments in R3 (varying lengths), we have A3 = Af x (0, oo).

(2.13.6)

IV By definition a flat in R3 is a bounded part of a plane. For simplicity we take flats to be discs of unit radius. Such a disc can be described by its center 9 e R3 and the direction co e E 2 normal to its plane. The space F of such flats can be represented as F = E 2 x U3. The result of §3.16, IV, can be considered as referring to triangular flats in (R3. A direct application of Lebesgue factorization yields table 2.13.1 for T-invariant measures on the above spaces

Table 2.13.1 Group

Factor measures

T2

Lebesgue measure on U; a finite measure on S t

T2

Lebesgue measure on U2; a measure o n § , x (0, oo)

Af = § 2 x U3

T3

Lebesgue measure on (R3; a finite measure on § 2

A 3 = S 2 x (0, oo) x [R3

T3

Lebesgue measure on U3; a measure on § 2 x (0, oo)

F = E 2 x f I3

T3

Lebesgue measure on IR3; a finite measure on E 2

Space Af = §! x

U2

^2 = 5 ! X (0, oo) x [R

2

36

2 Measures invariant with respect to translations

2.14 Product spaces with slits The translations of U2 or R3 induce transformations of the product spaces G x G, E x E, r x r . What can we state about translation-invariant measures on these product spaces? For simplicity we consider measures which are concentrated on the slitted versions Xl9 X 2 and X 3 of these spaces. I X 1 = ( G xG)\Z1? where Z x = {pairs of parallel or antiparallel lines}. A pair (gl9 g2) e Xx can be represented as follows: where & = gi(^g2 (the intersection point), and q>t is the direction of gt. Thus we have \

x

= {(cpu (p2): cpx / cp2 and
II X 2 = (E x E)\Z 2 where Z 2 = {(e1, e2)'. el9 e2 are parallel or antiparallel or intersect by a line parallel to the XOY plane} The line y = ex n e2 together with planar directions Ox and O 2 of the vectors normal to ex and e2 (these vectors lie in the plane orthogonal to y), determine the pair e1 and e2 completely. Thus ^2 = {(*i, ^2): #1 * ^2 and Ox # (D2 + 7r} x Txy. Using the representation of §2.7, III we find X 2 = {($!, O2): Oi ^ Q>2 and 2 + TC} X S 2 /2 X U2 = Y2 x 1R2. (2.14.1) Here we identify U2 with the xOy plane. Ill X 3 = (T x f ) \ Z 3 where Z 3 = {pairs of parallel or antiparallel lines}. For every (yl9 y2) e X 3 there exists a unique line segment s with endpoints on yx and y2 which has the shortest length among similar segments. Let

2.15 Almost sure T-invariance of random measures

37

Table 2.14.1

Space

Assume invariance of the measure \i

GxG

T2 (shifts of the plane)

ExE

Shifts of U3 parallel to xOy plane

li(Z2) = 0

Lebesgue L 2 on xOy\ a finite measure on Y2

r xr

T3 (shifts of U3)

fi(Z3) = 0

Lebesgue L 3 on U3; a measure on Y3 which is finite on every Y3(a)

Additional condition

Factor measures Lebesgue L 2 on U2; a finite measure on Yt

Q be the midpoint of 5, h be the length of s, CDi be the spatial direction of yi9 i = 1, 2. The parameters (Q, col9 CD2, h) uniquely determine any pair (yl9 y2) e X 3 . We have the representation X 3 = {(CDU CD2): CD1 ^ co 2 , CD1 ^

-CD2}

x {0, 00) x U3 = Y 3 x

U3.

We will also use the notation Y 3 ( f l ) = {(CO19 CD2)\ COt # CD2i CDX #

-CD2}

X ( 0 , a).

We conclude by Lebesgue factorization that any T2-invariant measure on Xx is a product of the Lebesgue measure L 2 and a measure on Yt. But, if a measure on Xx has to correspond to a measure \x on G x G for which //(ZJ = 0, then we must exclude the totally-infinite measures on Yx. Similar remarks are valid for other cases as well, and we obtain the factorization table 2.14A. We apply these factorizations in §2.16.

2.15 Almost sure T-invariance of random measures In 1968 a Cambridge mathematician RoUo Davidson (who died in an accident at the age of 26) wrote a remarkable thesis which dealt with line and flat processes. A phenomenon discovered by Davidson was the fact that, under rather broad conditions, the invariance of the first (/ix) and the second (ju2) moment measures of a random measure Y\ on the space G of lines implies that rj is invariant with probability 1 (i.e. almost sure). Davidson required invariance of fi1 and [i2 with respect to the (Euclidean) group of motions of the plane, but later this condition was replaced by T2-invariance. Davidson's original result ([43]) was extended in various directions by Krickeberg [8], Kallenberg [9], Papangelou [10], and Ambartzumian [34]. Below we treat this topic using the factorizations of §2.14.

38

2 Measures invariant with respect to translations

By definition, a random measure rj in some space X is a map rj: H -> JTX,

where O is a probability space and J^x is the set of measures on X. We introduce a natural cr-algebra on yTx a s the minimal one containing the sets {rjeJ^x:

rj(A) < x}

for the Borel i c X , and all x e U. As usual we require that the map r\ be measurable (i.e. for every Borel, A t](A) should be a random variable). The first and second moment measures of rj are said to exist if /*i = Eiy, \i2 = E(*y x rj) (where E stands for expectation) happen to be locally-finite measures (or simply measures in our terminology) in the spaces X and X x X, respectively.

2.16 Random measures on G Let rj be a random measure on G with T2-invariant moment measures fi1 and We will require additionally that (a) fi2(Zi) = 0, where Z x is the set of parallel or antiparallel pairs of lines in the plane. By virtue of (a) we can consider fi2 as essentially a measure on G x G \ Z l 5 and according to the first line of table 2.14.1 we have ^ 2 ( ^ 1 &Q2) = d^m(d(pl

dcp2),

where d ^ is the Lebesgue measure and m is some measure on Yx. We now apply the (' = cp p ' = p + u9 2

and let U be the group of 'independent shifts' of the factors in the product G x G (an element of (U2 is (ul9 u2)). The measure fi2 is invariant under transformations from U 2 . Proof The following formula is clear from fig. 2.16.1: d& = Isin^! - cp2)\~l dpx dp2. Therefore M2(d0i dg2) = I s i n ^ - cp2)\~lm{d(pl dcp2) dpx

dp2. 2

Clearly, this measure remains invariant under the group U . In particular for every Borel i c G w e have fi2(A x A) = fi2(uA x uA) fi2(A x A) = \i2{A x uA).

2.16 Random measures on G

39

Figure 2.16.1

These equations imply probability 1 invariance with respect to shifts of the cylinder. Indeed by the Schwartz inequality li2(A x uA) = Eri(A)ri(uA) ^

y/(Er1

(A)Erj2(uA))

2

2

= Erj (A) = ii2(A x A\ and, since here we actually have an equality, we conclude that, for fixed A and w, r](A) = c - rj(uA) with probability 1, c = c(A, u) is non-random. Progressing to expectations, we find T2-invariance implies w-invariance of fxx (by §2.9 and the Cavalieri principle). Thus c = 1. We have proved that for any fixed A c G and u f](A) = r] (uA)

with probability 1.

(2.16.1)

However, the subset of O, where (2.16.1) holds, may depend on A and u. Since they are more than countable in number, we cannot directly conclude that (2.16.1) holds for every A and u simultaneously (which is our aim for the moment). The remedy lies in considering appropriate 'dense' countable sets. We will use 'shields': a shield is a product of an arc from S l 5 and an interval from IR. Shields with rational vertices we will denote as A(r); similarly w(r) will denote rational shifts. It follows from (2.16.1) that, for any sequences {A^} and {u[r)}, P{fl(A\*) = i/(ii}rUir)), i = 1, 2,...} = 1. Given any two open shields A, A' a G such that A = uA (

we can find sequences {A p} and {w|r)} such that simultaneously A = lim A{[\

A = lim njrU{r)

40

2 Measures invariant with respect to translations

in the sense of monotone convergence. Then the equality rj(uA) = rj(A') = ^(A) will follow for all those realisations of Y\ for which holds, i.e. with probability 1. We have proved (2.16.1) for all open shields and all shifts u. By measure continuation we can conclude that (2.16.1) holds for any Borel A and all shifts u. This implies T2-invariance ofrj with probability 1. Applying Lebesgue factorization to § x x U9 we conclude that rj is probability 1, a product of a Lebesgue measure on U and a measure on S x . By the Cavalieri principle, rj is then T2-invariant (with probability 1). We summarize: Let r\ be a random measure on G = S x x U which has T2-invariant moment measures /i l 5 \x2. If condition (a) is satisfied then rj is T2-invariant with probability 1. It follows from the results of §2.8 that rj with probability 1 factorizes into a product of a Lebesgue measure on U and a finite random measure on S x . A similar probability 1 invariance-factorization result holds also for random measures on G.

2.17 Random measures on E Let rj be a random measure in the space E with T3-invariant first and second moment measures \ix and \i2. We assume additionally that \i2(Z2) = 0 see §2.14. Then rj is T3-invariant with probability 1 and therefore rj factorizes into a product of the Lebesgue measure on R and a random finite measure on S 2 . The proof follows the ideas of §2.16. We outline the main steps. Using the (cu, p) representation of the planes (see §2.3) we define a shift u of I as follows: co' = co p' = p + u.

Let d ^ be the Lebesgue measure on 1R2 in the factor representation (2.14.1). We have

d0>=f(couto2)dp1dp2, where / depends only on the orientations of the planes *i =(<*>i>Pi)>

e2 =

(co2,p2).

The determination of the exact form of the function/is left to the reader. This,

2.18 Random measures on F

41

together with the factorization of \x (see table 2.14.1), implies that \i2 is invariant with respect to independent shifts of the factor spaces in the product ExE. We then follow essentially the same steps as in the derivation of the previous section. A similar result also holds for random measures on E.

2.18 Random measures on F In this section we use the notation of §2.5 and §2.14, III. Let Y\ be a random measure on T possessing T3-invariant moment measures fi1 and \x2 (in particular, fi1 factorizes according to §2.9). We assume additionally that (a) ii2 (Z3) = 0. According to §2.14, (a) implies that A*2(dyi dy2) = d 6 m i ( d / l d c °i d(J°2l where dQ is Lebesgue on IR3 and mx is some measure on Y3.

(2.18.1)

If m1 factorizes into a product of Lebesgue measure on (0, oo) and some measure m2 on S 2 x § 2 , i.e. if m^dh dcox dco2) = dh m2(dco1 dco2)

(2.18.2)

then rj is T3-invariant with probability 1. The converse is also true. Sketch proof From (2.18.1) and (2.18.2) we have fi2(dQ dh dco1 dco2) = dQ dh m2(dco1 dco2).

(2.18.3)

Let us now fix the orientations co1 and a>2 of the lines. Let d^\ and d^ 2 be planar Lebesgue measures on planes through O orthogonal to the directions col and co2 (see §2.5). We have dQ dh = sin 6 dPx d^ 2 ,

(2.18.4)

where 6 is the flat angle between the planes carrying d ^ and d^ 2 . This result strongly depends on the fact that the segment h always remains parallel to the line of intersection of the planes containing d^\ and d^ 2 . Let us denote by u transformations of T which correspond to parallel shifts of U2 in the representation § 2 x R2 of F discussed in §2.7, III. The directions of lines under u remain unchanged. The measure m2(dco1 dco2) sin 6 d ^ d^ 2 does not change under application of independent transformations ux and u2 to the factor spaces in F x F. In particular for every A <= F we have

42

2 Measures invariant with respect to translations

\i2{uA x uA) = \i2(A x A), H2(A x uA) = fi2(A x A). Hence our assertion follows by essentially repeating the reasoning outlined in §2.16. The necessity of the condition (2.18.2) can be shown as follows. The probability 1 T3-invariance implies, according to §2.9, that Y\ factorizes with probability 1, i.e. Y\ 2

= L2 x v,

where L 2 is Lebesgue on U (and therefore non-random) and v is some random measure on S 2 . By the assumption (a), rj x rj will not charge the set Z 3 (probability 1). Hence we can use (2.18.4) and therefore the product measure rj x rj will possess a non-random measure factor d g dh. As a result, n2 will also have this factor. This completes the proof.

Measures invariant with respect to Euclidean motions

The requirement of invariance with respect to both rotations and translations of the basic space (i.e. invariance with respect to Euclidean motions) in fact enables us to determine measures in some of the spaces considered in chapter 2 in a unique way. Below we consider these questions, starting with a description of Haar measures on the groups of rotations of U2 and U3. To derive Haar measures on Euclidean groups (or kinematic measures) we apply the method of Haar factorization. A natural application of the factorization ideas leads to position-size-shape factorizations which we consider in the concluding sections. Some of these factorizations reappear in chapter 4 in a different context. There is a conceptual difference between the 'shapes' we consider in this chapter and the 'affine shapes' of chapter 4. Following the thinking which led to the term 'affine shape', the shapes of the present chapter should be termed 'Euclidean shapes'. However we cling to the shorter term 'shape' which is now widely used, [34], [41], [63].

3.1 The group W2 of rotations of U2 Rotations of U2 around the origin O can be represented by points on the unit circle § x : a point cp e Sx corresponds to the (anticlockwise, say) rotation by the angle cp. The product of rotations corresponds to addition (mod 2n). There is a natural measure on § : which ascribes to each arc a a S1 its length \a\. We call this the arc length measure and denote it by dcp. The arc length measure is invariant under rotations (because the lengths of the arcs remain invariant). We denote the group of rotations of U2 around 0 by W2, w e W2. By the correspondence

44

3 Measures invariant with respect to Euclidean motions

the arc length measure can be considered as a measure on W 2 . It is finite and both left- and right-invariant (the group W2 is commutative). From the remarks in §1.4 it follows that every Haar measure on the group W2 of rotations of U2 is proportional to the arc length measure.

3.2 Rotations of R3 We denote the group of rotations of U3 by W 3 . A rotation is a Euclidean motion of U3 which keeps the origin O intact. We describe two different representations of elements from W 3 . I Directed flags We call a figure which consists of a directed line y through the origin 0 and a halfplane h c IR3 bounded by y a directed flag (non-directed flags will be considered in chapter 5). An example of a directed flag yields the pair (y0, h0) which we describe in terms of the usual Cartesian coordinates in(R3: y0 - the axis Ox; h0 - the halfplane y > 0 of the z = 0 plane. Every w e W 3 can be completely described by the directed flag w(y0, ho\ i.e. by the image of (y09 h0) under w. In fact, given wy0 (the image of the Ox axis under w) we still have the freedom of rotation around the wy0 axis. We remove this freedom by fixing the position of wh0. Clearly the range of wy0 is the unit sphere S 2 ; with wy0 fixed, the range of wh0 is S x . However, an attempt to ascribe the topology of S 2 x Si to W3 fails (there will be pairs of flags near to each other both as sets and in directions of y and yet far apart as points in § 2 x S x ). We can represent a rotation as w = (O, 9\ where Q G S 2 is the spatial direction of wy0; 0> E U3 is a point on the plane t(Q) tangent to § 2 at Q; 0* belongs to the unit circle C(Q) c t(Q) centered at the point of tangency, 0> = whon C(Q). The topology in W3 is now defined by the convergence rule lim(Q n ,^) = (Q,^) if and only if lim QM = Q

in S ,

3.3 The Haar measure on W3

and

45

lim &n = 0> in (R3.

Actually W3 is a fibered space: TT(Q,

0>) = Q

G

§2;

S x is a fixed fiber model. Remark If we delete the fibers corresponding to Q1 - direction of the Ox axis, and Q 2 ~ the opposite direction, then this slitted W3 becomes identical to the product

{S2\{nl9a2})xsl9 i.e. we have W3 « § 2 x S x in the sense of §1.6. It is easy to complement this map to become (discontinuous) one-to-one. Then S 2 x S x becomes a measure-representing model of W 3 . (This follows from the fact that both spaces are compact.) Having this map in mind we will label rotations by points of S 2 x S l 9 i.e. we will use w = (Q,O),

Qe§2,

®G§!.

(3.2.1)

II Dual representation Parametrization of rotations by points from § 2 x S x can be achieved in a different way from (3.2.1) owing to the possibility of the dual description (co, cp) of a directed flag (y, h). Here 00 e §2 is a spatial direction defined by two conditions: (a) that CD is normal to the plane containing h; (b) an observer whose feet are at the centre of S 2 and whose head is at co looking in the direction of y will find h on his left hand. As soon as CD is determined, we define cp e S x to be the planar direction of y in the plane orthogonal to CD. In this way we get a parametrization w = (co, cp)

(3.2.2)

which we call dual to (Q,
3.3 The Haar measure on W3 We consider on the product space S 2 x S x the measure which is a product of the area measure on S 2 and the arc length measure on S x . This measure has two images on W3 generated by the maps (3.2.1) and (3.2.2). We naturally denote these images by dQ dO, corresponds to (3.2.1), and dco dcp, corresponds to (3.2.2).

46

3 Measures invariant with respect to Euclidean motions

Let us consider a map (Q, O) - w o (Q,
(3.3.1)

where w o (Q, O) is the result of an action o f w e W 3 on the flag (Q, is invariant under the transformation (3.3.1). (This follows from invariance of area and arc lengths under rotations.) But (3.3.1) corresponds to multiplication in W3: w o (Q, O) = vvvvj with wx = (Q, O). It follows that dQ dO is a left-invariant Haar measure on W3 with total value 2n - 4TT. By the remarks in §1.4 dQ dO is bi-invariant. A similar argument applies to the measure dco dcp. We come to the conclusion that dco dcp is also bi-invariant Haar. Since the total values of dQ dO and dco dcp coincide, we have by unicity dw = dQ dO = dco dcp,

(3.3.2)

a result often used in geometric probability [13]. It remains to add that the corresponding measure on W3 when viewed as a fibered space is a composition (see §1.6) of arc length measures on the circular fibers via the area measure on S 2 .

3.4 Geodesic lines on a sphere By 'polar mapping', any point coe S2 determines a directed geodesic line g on S 2 : by definition, the plane containing g passes through the center of S 2 and is orthogonal to co; the direction of g is chosen in such a way that it is seen anticlockwise from the point co. Therefore the space Go of directed lines on § 2 can be identified with S 2 itself: Go = S 2 .

Let us denote by n the measure on G o which is the image of the area measure dco under the polar mapping. Clearly fi is invariant under rotations of § 2 (because dco is), but is \i unique? This question can be given an affirmative answer by using the constructions of §3.2. Let us consider the space G o x § x of pairs (g, cp) = (co, cp), where g e G o and cp is a point on g. Every (g, cp) can be identified with a rotation (see §3.2) and therefore there is only one finite measure on the space G o x § ! which is invariant under W3, namely dco dcp.

Let m be any measure on G o which is finite and invariant under W3. On the space G o x S x we consider the product measure m(dg) • dcp.

3.6 The invariant measure on G and G

47

By the Cavalieri principle, this measure is W3-invariant. Therefore, m(dg) dcp = c dco dcp for some c ^ 0. Eliminating the measure factor dcp, we find that m(dg) = dcoc. This means that m is necessarily proportional to /j. Note that we have also proved the uniqueness of the measure dco.

3.5 Bi-invariance of Haar measures on Euclidean groups We denote by M n the group of all Euclidean (rigid) motions of Un (we actually consider the cases n = 2,3), M e M n. Let us see how the criterion of §1.3 applies to Mn. The subgroups (U and V are U = ¥„, the translations of Un and V = WB, the rotations of Un. Both representations of (1.3.4) are known to exist; hv is the Lebesgue measure (also denoted as dt); hv is the Haar measure on Wn (also denoted as dw). The equation r,wr = w,tr has the following solution: (3.5.1) Wr = Wj.

The above is a Cavalieri-type transformation of Wfl x Tn. Indeed, for each wl9 the transformation given by (3.5.1) preserves the Lebesgue measure on Jn (this follows from rotation-invariance of Lebesgue measure in Un). Thus dM = dtldwT=dtTdwl

(3.5.2)

is a bi-invariant Haar measure on Mn. In the sections that follow we also denote by Mn the groups of transformations of the spaces of lines and planes in Un (n = 2, 3) induced by the corresponding Euclidean groups.

3.6 The invariant measure on G and G Any M 2 -invariant measure \i on G = S x x U is proportional to the product measure / x Ll9

(3.6.1)

48

3 Measures invariant with respect to Euclidean motions

where / is the arc length measure on S x and L x is a Lebesgue measure on U. We will also denote the above measure by dg. Thus, in the coordinates introduced in §2.1, dg = d(p dp.

(3.6.2)

To prove the assertion we consider a product set

AxB

with A

^§UB^U.

Since T2 is a subgroup of M 2 , \i necessarily must factorize as in §2.9, i.e. H(A x B) = m(A)-L1(B).

(3.6.3)

Let us now apply a rotation w to A x B. We have w(A x B) = wA x B. Taking the measure \i from both sides and applying (3.6.3) we find that necessarily m(wA) = m(A), i.e. m should be invariant with respect to rotations, i.e. up to constant factor m = l. It remains to check that the measure dg is indeed invariant with respect to M 2 , but this follows from the possibility of presenting any M e M2 as a product of a parallel shift and a rotation. Using the factorization described in §2.9 it is also easy to prove that any M 2 -invariant measure on G corresponds to a product measure on the model E x x U of this space (see §2.7). The factor measures are (up to constant factors) on E x - the arc length measure dcp, on U - the Lebesgue measure dp.

3.7 The form of Ag in two other parametrizations of lines Since we have dg in (cp, p) coordinates, the form of dg in terms of other parameters can be found by Jacobian calculations. However, in the following we use the more geometrical 'symmetry principle'; the latter is especially useful when the dimensionality of the problem increases (see §3.10). We consider the following: (a) The (ij/, x) coordinates as described in §2.7, 1. They are applicable because the invariant measure of G\G X is zero. (b) The (ll912) coordinates. Let us assume that two 'reference lines' gx and g2 are fixed on U2. Let lh i = 1, 2, be the usual one-dimensional coordinates on gt.

3.7 The form of dg in two other parametrizations of lines

49

Figure 3 J.I

We can exclude lines parallel either to gx or to g2. On the remaining set, the points of intersection completely determine the line g (fig. 3.7.1). Let us write using (a) dg = Fdldil/.

(3.7.1)

Our problem is to find that particular F which yields the required invariance property. Geometrically, the element F d/ dij/ has the meaning of the value of the measure of set of lines {g e G : g intersects an interval (/, / + d/) c g0, the intersection angle \jj lies in (ij/, i// + d\j/)}. We choose two arbitrary linear elements, dlx and d/2, on U2 (necessarily on different lines but of equal lengths). The corresponding sets (3.7.2) (with the same \j/, \jr + d^) will be congruent. From this we conclude that (1) the function F in (3.7.1) is universal, i.e. it does not depend on the choice of reference line g0; (2) F does not depend on /, i.e. F = F(iA). To determine F let us choose a pair of lines g1 and g2 and elementary intervals d'i ^ Gi a n d d/2 cz 0 2 , and let A = {g e G : g intersects both d/x and d/ 2 }. By first choosing g± to be the reference line in the representation (a) we write (fi is another notation for the measure d#)

50

3 Measures invariant with respect to Euclidean motions

Figure 3.7.2 p is the distance between d/x and d/2 In the notation of fig. 3.7.2 sin \j/2 P

Thus This expression should be symmetrical in the indices 1 and 2 (since we could start the derivation from g2). The variables i//1 and \//2 are independent. Therefore we have to conclude that F(ij/) = c - sin \ff. The calculation of the measure of lines which hit a unit disc (say) shows that, in order to get dcp dp, we have to choose c = 1. Thus we have found the following expressions: dg = sin i// d/ dxj/

(3.7.3)

and dg = p'1 sin \j/1 sin i//2 d/x d/2.

(3.7.4)

Remark The above can be considered as an independent proof of the uniqueness of invariant measure on G, in the class of measures possessing densities of the form (3.7.1).

3.8 Other parametrizations of geodesic lines on a sphere Here we consider the space G o of non-directed great circles. (a) Let us assume that a 'reference' great circle g0 is fixed on S 2 and let g$ denote a semicircle in some way specified on g0. If we ignore the geodesies which pass through the endpoints of #Q, then every g e G o is determined by the unique point / of its intersection with g J and by the

3.9 The invariant measure on r and r

51

angle i// of intersection of g with g J at /. Thus flf = (/,^),

/e(0,7r),

^e(0,7c).

(b) Let two 'reference' great semicircles gfj" and g2 which lie on different great circles be fixed on S 2 . Except for the geodesies which pass through the point g\ n g2 each g e G is determined by the pair (Zl5 / 2 ), where lt is the intersection point lt = g n #*. Thus, g = (lul2%

each/,-e (0,7c).

The form of the invariant measure d# in the coordinates (/, if/) and (/1? l2) can be found by Jacobian calculation starting from do; = sin 6 d6 dO,

(3.8.1)

which is the usual expression of an area element in polar angular coordinates on S 2 . However, an approach in the style of §3.7 (the symmetry principle) is also possible. The result is dg = sin \// Al di//

(3.8.2)

and dg = (sin p)" 1 sin xfj^ sin \\i2 d\x d/2.

(3.8.3)

Here d/x and d/2 are length elements, p is the geodesic distance between dlx and d/2, and ^ is the angle of intersection of g* by g.

3.9 The invariant measure on T and T Any M3-invariant measure fi onT is proportional to the product measure a x L2,

(3.9.1)

2

on the product model S 2 x IR of the space Y where a is the area measure on § 2 , and L 2 is the Lebesgue measure on U2. To obtain the proof we consider the set A = {yeT'.neB,^

belongs to the unit disc Kx centered at 0}

(see §2.5 for notation). For every rotation w e W3, wA is a product set wA = wB x Kx. Therefore, by the factorization result in §2.9 concerning T3-invariant measures onf, amounts to W3-invariance of the factor measure on S 2 . Therefore the latter measure is necessarily proportional to the area measure a (the uniquess of a was essentially shown in §3.4). Finally y 3 -invariance of (3.9.1) can be checked by applying the Cavalieri principle (see §1.5).

52

3 Measures invariant with respect to Euclidean motions

A similar result holds for the space F, in which case the area measure a is defined on E 2 (see §2.7). In terms of the parametrization described in §2.7 the measure (3.9.1) is written as dy = dQ d0>9

(3.9.2)

where dy is an element of/i, dQ is an element of a, and d£P is an element of L 2 . The representation (3.9.2) is also valid for the M 3 -invariant measure on the space F.

3.10 Other parametrizations of lines in U3 Below we consider the form of dy in two other parametrizations of lines from r . (a) Let e0 be a 'reference plane' fixed in U3. If we ignore lines parallel to e0 then any line y can be described as

y = (ft Q\ where Q is the point in which y hits e0, and Q e S 2 /2 is the direction of y (compare with §2.7). (b) Let us assume that two distinct reference planes e1 and e2 are fixed in IR3. Ignoring those lines which are parallel either to ex or to e2 or hit the line gj n e 2 we can write 7 = {6i> 62} ( a n unordered pair), where Qt is the point where y hits e{. Applying the symmetry principle (similar to that used in §3.7) requires the consideration of infinitesimal flats rather than segments. However, the main idea is the same. The result is as follows: dy = cos v dQ dQ,

(3.10.1)

where dQ is an element of area measure on S 2 , dQ is an element of planar Lebesgue measure on e09 v is the angle between the direction Q and the direction normal to e0, and

where vf and dQt have similar meanings with respect to eh i = 1, 2, and p is the distance between dQx and d<22The basic property of !R3 which is responsible for the final form of the above expressions is the formula giving the solid angle dQ at which an infinitesimal flat of area dQ is seen from a point distance p apart, namely dQ = p~2 cos v dQ, where v is the angle between the direction of p and the direction normal to dQ.

3.12 Other parametrizations of planes in R3

53

3.11 The invariant measure in the spaces E and E Any M 3 -invariant measure \i on E = S 2 x U (see §2.3) is proportional to the product measure axL, (3.11.1) where a is the area measure on § 2 , a n d ^ i is t n e Lebesgue measure on U. The proof is similar to that used in §3.6 and §3.9 (i.e. using factorization described in §2.9 and the uniqueness of a on S 2 as a W3-invariant measure). A similar result holds for E in terms of its product model E 2 x R (see §2.7; the measure a is well-defined on E 2 ). In both the spaces E and E the element de of the invariant measure is written in the form de = dcodp, (3.11.2) where dco is an element of a, and dp is an element of Lx.

3.12 Other parametrizations of planes in U3 Below we consider the form of de in three other parametrizations of planes. (a) Let us assume that a reference axis y0 is fixed in IR3. If we ignore planes parallel to y0, then any plane e can be described as e = (co, x\ where x is the point where e hits y0, and co is the direction normal to e. (b) Let us assume three distinct axes yl9 y2 and y3 are fixed in IR3. Ignoring planes parallel to the axes, as well as planes through points of their intersection (if any exist), we can write e=

(xl9x29x3)9

where x( is the point where e hits yf. (c) Let us assume that a reference plane e0 is fixed in R3. Ignoring planes parallel to e0 we can write

e = (&
(3.12.1)

where v is the angle between co and the direction of y0, dx is a length element, and dco is an element of area measure on S 2 ; =

cos vt cos v2 cos v3 2JA|

*

2

3

'

(3.12.2)

54

3 Measures invariant with respect to Euclidean motions

Radial projection of dy3 Figure 3.12.1

where vt is as above but referred to yi9 and | A| is the area of the triangle A which has its vertices on each of the elements dx f . The two above formulae are related by the value of da>1? which is the solid angle subtended by the normals to the planes from the set {the planes through an endpoint of dx 1? which hit both dx 2 and dx 3 }. We have 1=

cos v2 cos v3

2JAJ

2

n

3

*

n

.

(3.12.3)

In proving (3.12.3) the formula (3.8.3) can be useful. Indeed, the value of the solid angle in question will not change if we replace dx 2 and dx 3 by their projections dy2 and dy3 on the lines which emerge from the corresponding vertices of A perpendicularly to the plane of A. The length of dyt is cos vt dxh i = 2, 3. Next we note that da>1 equals the invariant measure of the set of geodesic lines which hit both radial projections of dy2 and dy3 (see fig. 3.12.1) on the unit radius sphere centered at dx1. The length of these projections is dyt fcj"1, where h{ is the distance between dx x and dx/5 i = 2, 3. Since our radial projections are perpendicular to the geodesic line joining them, (3.8.3) yields 1

cos v2 cos v3 h2h3 sin p

2

3

'

which is the same as (3.12.3). From (3.12.2) the form of de in the parametrization (c) can be derived in the following way. Let us assume that all three segments dx 1? dx 2 and dx 3 are parallel, and that both dx x and dx 2 lie in e0 and are both orthogonal to a line g joining them. Let i// be the flat angle between e0 and a plane through dx 1? dx 2 and dx 3 ; let the distance between dx x and dx 2 be p\ and let the distance from g to

3.13 The kinematic measure

55

dx 3 be h. Then (3.12.2) yields de =

sin3 il/ dxi dx 2 dx 3 . ph

Using

(see §3.7) and dx3

dij/

h

sin \jf

we find de = sin 2 ^ dg dij/.

(3.12.4)

3.13 The kinematic measure Let J be a 'figure' in (Rn. We denote by MJ the result of applying the motion M e Mn to J. We will consider the space J of figures congruent to J, i.e.

Examples (1) (2) (3) (4)

J is a ball in R". Then J = R". J is a directed segment in (R3. Then J = A? (see §2.13). J is a directed segment in U2. Then J = M 2 . J is a pair of non-collinear segments of different lengths emerging from 0. Then J = IU3.

The measure on J which is the image of the Haar measure on M „ under the mapping

M-+MJ is called kinematic. Thus, the kinematic measure essentially differs from dM only in the cases where the above mapping is not one-to-one (as in the case in Examples (1) and (2)). In many cases the results of the previous sections imply the following proposition. Any measure on J which is invariant with respect to the group MM is proportional to the kinematic measure. The proof is tautological whenever J = Mn. Let us outline the proof for the J of Example (2). Let \x be an M 3 -invariant measure on Af. By the Cavalieri principle, the product measure \i(db) dO on the space

56

3 Measures invariant with respect to Euclidean motions

\* x S x is invariant with respect to M3. But the above product space is a measurerepresenting model of the group M 3 (the corresponding construction is left to the reader). Any M3-invariant measure on this model is necessarily proportional to the Haar measure on IU3 (by the uniqueness of the latter). Using the results of §3.5 and §3.2 we conclude that fi(d3) dQ = c dt dQ dd>, where c is a constant. Denoting the kinematic measure in the case c = 1 as d<5, we get by integration dS = 2ndtdQ.

(3.13.1)

Since every d e A* can be described by its continuation y eT and the onedimensional coordinate x of its endpoint on y, we have

A? = F x R. The kinematic measure on AJ is In times the product of the invariant measure on r and the Lebesgue measure on U9 i.e. dd = 2ndydx.

(3.13.2)

Proof By the Cavalieri principle, dy dx is y 3 -in variant and is therefore proportional to d<5. It remains to check that the proportionality constant is 2TC. There is a similar relation in the space Af. We recall that AS = G x R = M2 (see §2.13 and notation therein). We have dM = dd = dgdx.

(3.13.3)

The same idea can be used to derive a useful representation for dM, the element of the Haar measure on M3. Let sx and s2 be the two segments emerging from 0 see (4) above. The image of the two segments under M e M 3 defines M completely. We can describe Msx and Ms2 by determining the plane e where they lie and by their position k on e. Let d/c be the planar kinematic measure. By the Cavalieri principle, de d/c is M 3-invariant and is therefore proportional to the Haar measure dM on M 3 . A simple check shows that for dM as in (3.5.2) the proportionality constant is 1. Thus we have dM = dedk

(3.13.4)

which is an analog of (3.13.3). Along similar lines we briefly derive one more useful result. Every M e M 3 can be described by the parameters M = (e, g, x),

3.14 Position-size factorizations

57

where e is the image of the xOy plane under M, g is the image of the x axis (therefore g c: e\ and x determines the position of MO on g. Because (g, x) can be identified with k (a position on e\ from (3.13.3) and (3.13.4) we find dM = dedgdx.

(3.13.5)

On the other hand, we can also describe M as M = (y, O, x), where y e T is the image of the Ox axis, O e S x is the angle of rotation of e around y, and x determines the position of MO on y. By a kind of Cavalieri principle the product measure dy dO dx is a left-invariant Haar measure o n M 3 . From the uniqueness of the latter we conclude that dy dO dx = c • dM, or, using (3.13.5), cde

dg dx = dy dQ> dx.

Calculation of the measure of the set {M : MO belongs to the unit ball} shows that c = 1. Elimination of the dx factor yields dedg = dyd®.

(3.13.6)

We have obtained two expressions of the (unique) M^-invariant measure in the space of figures J — (a plane e; a line g on e). Strictly speaking, this measure cannot be termed kinematic; indeed, the image of dM in the space of the above figures is not locally finite. We use (3.13.6) in §6.3.

3.14 Position-size factorizations In this section (and in the next two) we consider products of the M -invariant measures that we met in previous sections. It follows from our factorization proposition of §1.3 that these products always allow separation of the kinematic measure (i.e. the unique M -invariant measure in the space of'positions' of the geometrical figure in question). We are interested in the measure factor that remains; in this section we consider cases where this factor is a measure in the 'size' space (which is always (0, oo)). Below we write the product measures in differential form and derive the desired factorizations using simple 'chain' procedures. Justification of each step in these chains can be based upon the following two rules:

58

3 Measures invariant with respect to Euclidean motions

(1) The Jacobians of transformations of the form yi = /i(xi,...,x m ) and yx =/ 1 (x 1 , ..., /»(xi,..., xj =

x

ym = fm(xl9...,

xj

m+l

coincide. This means that, where only a part of the variables are transformed and others are left intact, we have a gain in dimensionality. It is easier to calculate several Jacobians of reduced dimensionality than one of full dimension. (2) The invariance properties of the measures yield 'universality' (see §3.7 for an example) of their differential representations. Therefore the applicability of the latter is not influenced by the dependence of 'reference' elements on parameters. The benefit is twofold: we apply ready differential formulae from the preceding sections: we also achieve in this way 'explanations' for the resulting densities. I Pairs of points in U2 An ordered pair (Ql9 Q2) with Ql9 Q2 e R2, Q1 # Q2, can be represented by a segment S from A2. Using (2.13.3) we get {QuQi) = {MJ\ where MeM2J e(0, oo) (/ is the distance between Qx and Q2). Thus (see §1.6) U2 x IR2 « M2 x (0, oo). We denote by dQf planar Lebesgue measure elements. The measure dQx dQ2 in IR2 x U2 is M 2 " mvar iant and therefore necessarily admits Haar factorization (or, equivalently, separation of kinematic measure). If we take dQx dQ2 to be as shown infig.3.14.1 we will have

Figure 3.14.1 dQt, i = 1, 2, arerightangles, the segment /joining them is perpendicular both to dlt and to d/2

3.15 Position-shape factorizations

59

dQi dQ2 = d/x d/2 dtx dt2 = I dg dtl dt2 = ldgdt d/ = dM/d/.

(3.14.1)

Also we used (3.7.4) (with ij/1=il/2 = n/2) and (3.13.3). Also

t = ±d + hi where tx and t2 are one-dimensional coordinates of Qx and Q2 on the line joining these two points, dM is the Haar measure on M 2 (or the kinematic on Af). (3.14.1) is the desired factorization. II Pairs of points on the unit sphere Similar arguments remain valid when we consider the pairs (col9 co2) of points on S 2 , the unit sphere. Following (3.8.3) dco1 dco2 = dQ dcpi dcp2 sin p = sin p dw dp, (3.14.2) where dcDt are elementary solid angles, Q is the direction perpendicular to both co1 and co2 (dQ is the corresponding elementary solid angle), p is the angle between co1 and a>2, cpx and cp2 are the angles determining the position of co1 and co2 on the geodesic through col9(o29 and dw is an element of Haar measure on W3 (see §3.3). Strictly speaking, p cannot be called a 'size' parameter. III Pairs of points in U3 Similarly, an ordered pair (^\, ^ 2 ) , ^ \ ^ ^ 2 , of points from U3 can be represented as (3,1) with 3 e A?, / e (0, oo) (/ is the distance between ^ and ^ 2 ) . Thus [O3 w O 3 ^j

A •

vv (f\

_^-K\

We now denote by d^f the elements of Lebesgue measure in U3. The product measure d^\ d^*2 is M3-invariant, therefore it admits separation of the kinematic measure in Af (see §3.13). Using (3.10.2) and (3.13.2), by transformations similar to those in I, we get the desired position-size factorization d^ x d^ 2 = - 1 - d(512 d/.

(3.14.3)

2TT

3.15 Position-shape factorizations In this section we continue with separation of kinematic measure from products of M-invariant measures (see the introductory remarks to §3.14). In the cases we consider here the non-kinematic factors are measures in different 'shape' spaces.

60

3 Measures invariant with respect to Euclidean motions

I Pairs of lines on U2 On G x G we consider the product measure dgx dg2, where the dgt are elements of the iy 2 -invariant measure on G (see §3.6). It is not difficult to justify the use of (3.7.3) for dg2, where we take g1 for the reference line, i.e. (see fig. 3.15.1) d#i dg2 = dgx |sin \j/\ dty dx. Writing, according to (3.6.2), d
(3.15.1)

II Pairs of planes in U3 On E x E w e consider the product measure dex de2i where de{ are elements of the M 3 -invariant measure on E (see §3.11). Using ex for the reference plane, we put de2 in the form (3.12.4): de x de2 = dex sin 2 \j/ dg dij/, where \// e (0, n) is now the flat angle between ex and e2, and dg is the invariant measure of directed lines on e1. Each pair (e, g) can be represented as (y, ), where y e F coincides with g (viewed now as a line in IR3), and the angle O e S x determines the rotation of ex around y. From (3.13.6) dex de2 = dy d
(3.15.2)

This is the desired (position-shape) factorization. Indeed, y and O together determine the position of (eu e2) in IR3, and dy dO is essentially the only M 3 -invariant measure in the space of these positions. The shape parameter is the angle \j/.

Figure 3.15.1

3.16 Position-size-shape factorizations

61

III Triads of planes in IR3 We use the same notations as in II above. On E x E x E we consider the product measure dex de2 de3. Using ex for the reference plane, we write, using (3.12.4), de1 de2 de3 = del sin 2 fa &9i

sm2

^3 dfif3 d\j/2 d\j/3

where gt is the directed line of intersection of et with el9 i = 2, 3, and ^ is the corresponding flat angle. Using (3.15.1) we find de1 de2 de3 = dex dM |sin a| da sin2 \j/2 d\j/2 sin2 ij/3 di^3,

(3.15.3)

where a denotes the angle between the lines g^ and g2. This is the desired position-shape factorization, since, according to (3.13.4), de dM (where dM stands for the Haar measure on M 2 ) gives the Haar measure on M 3 .

3.16 Position-size-shape factorizations This section continues the work of the previous two (see the opening paragraph of §3.14). Here the non-kinematic factors are measures in 'size-shape' spaces. This factor permits further factorization, which results in the separation of measures in 'size' and 'shape' spaces. Although in each case the measure in the size-shape space is unique, the factor measure in the shape space may depend on the choice of the 'size' parameter. I Pairs of lines in U3 We consider the product measure dyx dy2 on the space F x r , where dy is the M 3 -invariant measure on F. Our aim is to obtain the position-size-shape factorization dy1 dy2 = dM • m(dh)ii{do\ where dM is the Haar measure on M39 and h and a are parameters describing the size and shape of a figure yl9 y2. Here h is the (minimal) distance between the lines, and a is the angle between the spatial directions co1 and co2 of the lines. We start with the representation (see §3.9) dy, = d(ot d&h

where dco£ is the solid angle element and d^t is an area element on a plane perpendicular to y(. We have d?i dy 2 = dco1 da>2 d ^ \ d&2 = dQ sin a dcpt dcp2 d0*x d&2.

Here Q is the direction perpendicular to both a>1 and co2 and cpl and cp2 are one-dimensional angles determining the positions of a>1 and co2 on the plane perpendicular to Q (see (3.14.2)). We can represent each d^t as a product of

62

3 Measures invariant with respect to Euclidean motions

two linear elements, one of which (dxt) is directed along Q, while the other (dpi) lies in the a>l9 co2 plane and is perpendicular to (D{. The product dpt &(p{ = dgt

yields the element of invariant measure of lines on a plane (see §3.6). Therefore,
(3.16.1) 2

i.e. the distribution of the angle o is proportional to its sin , the measure in h is Lebesgue. II Triads of lines in U2 If the case of parallel lines is ignored, then each ordered triad of lines (01,02*03) generates a labeled triangle in R2 (in a labeled triangle the sides (or the vertices) are given the numbers 1, 2, 3). Our purpose now is to obtain a position-size-shape factorization of dgx dg2 dg3. First we look at the space £ of (labeled) triangular shapes. Suitable coordinates in the space £ can be as follows. We fix a 'base' a of unit length on the plane and put two lines through the endpoints of the base under angles £)l and £ 2 , as shown in fig. 3.16.1. Except for the case of parallel lines, our construction always defines a labeled triangle. However, £x and ^ 2 do not always coincide with the interior angles ax and a 2 of the triangle.

Figure 3.16.1 The numbering of the angles corresponds to the numbering of the vertices. In the usual numbering of sides a = a3

3.16 Position-size-shape factorizations

63

We have £l=01>

^2 = «2

if^ 1 + ^ 2 < 7 C ,

and £1=n-a1,

£2 = n-(x2

if ^ + f2 > rc.

The two cases are related via reflection. Anyhow, for our purposes, the space of triangular shapes can be identified with the product E « (0, n) x (0, n). For a formal definition of a triangular shape and a complete description of the space E, see §3.18. However, the above product description will be acceptable until we consider measures on £ given by densities /({i,{2)d{id{2. Returning to our problem, we take g3 for the reference axis and use (3.7.3) to get dg1 d# 2 d# 3 = d# 3 sin ax sin a2 d£x d£ 2 dxx dx 2 . Clearly dxx dx 2 = dt da 3 , where t stands for a shift along the g3 line, a 3 is the length of the side of the triangle formed by the lines gl9 g2 and g3 which lies on g3 and dt and da3 are the elements of Lebesgue measure in the corresponding spaces. By (3.13.3) d0! dg2 dg3 = dM da3 mfl3(d
(3.16.2)

where the Jacobian J is written in terms of the interior angles (seefig.3.16.1) as J =

sina 3 sin ccl + sin oc2 •+• sin cc3

and therefore m

sin <*! sin a2 sin a 3 (da) = — r=-— d£x d£2. *v s i n a 1 + s i n a 2 + sina 3

/^i^^ (3.16.3)

III Triads of points in R2 We denote by dQt the Lebesgue measure elements in U2. The previous results can be used to obtain position-size-shape factorizations of dQx dQ2 d g 3 .

64

3 Measures invariant with respect to Euclidean motions

Figure 3.16.2

It is known that [2] dQi d g 2 dQ3 = 8r3 dgx dg2 dg3,

(3.16.4)

where r is the circumradius of the triangle QiQ2Q3 while gl9 g2 and g3 are (continuations of the) sides of the same triangle. Let us give a simple derivation of this relation. Using (3.14.1) we find d#i dg2 dg3 = sin \j/1 dxx dij/1 sin \\i2 dx2 d\j/2 dg3 = sin i//1 sin \\i2 d\\i± d ^ 2 ( a 3 ) - 1 dQ1

dQ2.

As shown in fig. 3.16.2, d<23 = sin i/^3(sin ij/3)~1a1 d^ 2 (sin il/3)~1a2

d\j/1,

therefore Now (3.16.4) follows by substitution since sin ^ ! sin \\i2 sin 'A 3 (^i^ 2 ^ 3 )~ 1 = (2r)~3. It is a matter of basic geometry to express 8r3 in the form 8r 3 = /i3(sin ax + sin a 2 + sin a 3 )" 3 . Hence, using (3.16.2), we find dQi d<22 dQ 3 = dM h3 dh vh(d
(3.16.5)

where the measure vh in the space £ is given by the density _

sin a t sin a9 sin a, _ ,„ (sin a i + sin a 2 + sin a 3 ) 4 ^ ^* From an elementary relation between h (the perimeter) and S (the area) of the triangle QiQ2Q3, which is as follows: rtV

x 7

2

we find that

sin2 a t sin2 a 2 sin2 a 3 4 4 (sin OL1 + sin a 2 + sin a 3 )

3.16 Position-size-shape factorizations

65

vs(dor) - (sin a1 sin a 2 sin o^)"1 d ^ d£ 2 ,

(3.16.7)

which appears in the position-size-shape factorization

f] dQi = 2dM SdS

(3.16.8)

Lastly we write the corresponding density of shapes when A, the area of the circumcircle through Ql9 Q2, 6 3 , is chosen for the size parameter. Since A = —(sin ax sin a 2 sin a 3 )

1

S9

we have i dQ2 dQ 3 = -~2 dM A dA sin ax sin a2 sin a 3 (3.16.9)

= dM AdA vA(da) with vA(da) = —~ sin ax sin a 2 sin a 3

(3.16.10)

7T

IV Triads of points in U3 Three points Ql9 Q2, Q3 e IR3 can be described by means of the plane e through these points and the positions ^ , 0>2 and ^ 3 of the points on e (thus ^ are points in two dimensions). We choose the elementary volumes dQu dQ2 and dQ 3 as shown in fig. 3.16.3, so that d& = dJ,d£l

(i=l,2,3).

In this situation (see (3.12.2)) d^ = (25" 1 )d/ 1 d/ 2 d/ 3 , where S is the area of the triangle Thus

9^2»z.

dQi dQ2 d g 3 = 2 de S d&x d&2 d0>3. Factorizing d^ x d^ 2 d^ 3 according to (3.16.8) and making use of (3.13.4), we find that dQi dQ2 dQ3 = dM S2 dS vs(dcr),

d/,

Figure 3.163 d^f lies on e, dlt is perpendicular to e, i = 1, 2, 3

66

3 Measures invariant with respect to Euclidean motions

where &M is the Haar measure on M 3 and the measure vs on £ (the space of triangular shapes) is the same as described in subsection III. Note that the measure in the space of areas differs from that in (3.16.8). V Quadruples of planes in U3 The problem here is to factorize out the Haar measure on fV03 from de1 de2 de3 de4; we use the same notation as in §3.15, III. Choosing ex for the reference plane and applying (3.12.4) we get 4

Y\dei = de1 Y\ sin2 ^ d^dgt. 1

2,3,4

Further application of (3.16.2) and (3.13.4) yields nde-dMd^m^d^) n sin2^d^, 1

(3.16.11)

2,3,4

where hx and cr1 denote the perimeter and the shape, respectively, of the (triangular) face of the tetrahedron built by the four planes which lies on the e± plane. Equation (3.16.11) solves our problem; its asymmetry is due to the choice of the size parameter hx. In principle this can be remedied by moving to symmetrical size parameters (like the volume of the tetrahedron, etc.). VI Quadruples of points in (R3 Of more importance in the context of the present book is the shape measure which is obtained from dQx dQ2 dQ3 dQ4 (the product of Lebesgue measures in R3) by separation of the kinematic measure and a measure in the space of size parameter V, the volume of the tetrahedron 61626364In U3 we consider the usual polar coordinates with origin at 61 • In particular let r2, Q e § 2 be the polar coordinates of 62 s o that d6 2 = r\ dr2 dO. We denote by v3 and v4 the angles between Q and the directions of 6163 and 6164J w e denote by O the rotation angle of the plane 616263 aroundft;and we denote by O 4 theflatangle between 616263 and 616264- In this notation ^63 = rl dr3 sin v3 dv3 d® and ^64 = r4 dr4 sin v4 dv4 dO4. Using (3.5.2) and (3.3.2) we find n d 6 f = dM i=l

[ ] rfdri \\ sin v, dv£ dO 4 . i=2,3,4

i=3,4

It is not difficult to derive fromfig.3.16.4 that V = ^r2r3r4 sin v3 sin v4 sin O 4 = ^rf a • /? sin v3 sin v4 sin O 4 ,

(3.16.12)

3.17 On measures in shape spaces

67

QA

Figure 3.16.4

where a = r 3 /r 2 ,

p = rjr2.

The replacement in (3.16.12) of the variables r2, r3 and r4 by F, a and /? yields 4

[ ] dQi = dM V2 dV m F (da 1 ),

(3.16.13)

where the measure mv in the space of tetrahedral shapes ax has the form mF(dcr1) = 6 3

~ , 4 —rA . z a • p • sin v3 sinz v4 sin J O 4

(3.16.14)

3.17 On measures in shape spaces All measures in the shape space T, derived in the previous two sections, except for the measures (3.16.7) and (3.16.14) are finite (and therefore can be normalized to become probability measures). A less obvious case is the measure mh given by (3.16.3). Let us prove that it is finite. We take a product set A x B x E, where A is a bounded set in the space of positions of triangles; for instance, A = {the center of gravity of the triangle formed by gl9 g2, g3 lies in the unit disc} B is a bounded set in the space of perimeter lengths, say B = (0, 1) In terms of the product cylinder topology (see §2.1) the set A x B x L has compact closure (because the distances from 0 of the lines in (gl9 g2, g3) e

68

3 Measures invariant with respect to Euclidean motions

i x B x L remain bounded). Thus the value of the measure dg1 dg2 dg3 on this set is finite. By factorization it follows that mh(L) < oo. In §6.11 we give the values of mh on a family of subsets of 2; in particular ^-=^.

(3.17.1)

By a similar argument we can show that the measure (3.16.6) is also finite. Again, values of the measure vh (as well as of the measure v^) on a family of sets can be found in §6.11. The following important observation is due to Miles [57]: the shape density (3.16.10) survives for triads of points on a sphere in M3. Let Ql9 Q2 and Q3 be three points and let dg, (i = 1, 2, 3) be area measure elements on the surface of a sphere of radius R. The problem is to express the product measure dQ1 dQ2 dQ3 in terms of the parameters w-rotation of the sphere, r-the radius of the circular trace of the sphere on the plane through Qu Q2 and Q 3 , cr-the planar triangular shape generated by Ql9 Q2, Q3. We can choose dQi = r dpi dx( where dft are angles which correspond to elementary arcs on the circle through Ql9 Q2, Q3, the linear element dxt is perpendicular to d $ and lies in the tangent plane at Qt. Using (3.12.2), (3.11.2) and (3.3.2) we obtain d £ i dQ2 dQ3 = r 3 2|A|(cos v)" 3 de dft dj?2 dp3 = r 3 2|A|(cos v)" 3 dw dp d&> dp3 Observing from (3.16.10) that we easily transform the above to the result of Miles dQi d g 2 dQ3 = In2 dw R 3 r 3 [l/ V / (R 2 - r 2 )] dr vA(do). Apart from the result we present in §6.11 this can be used in calculation of vA(H). We integrate both sides over the cube of the sphere and obtain

(4nR2)3 = 2n2'4n'2nR3

CR Jo

r3[\/j{R2

- r 2 )] dr-vA(L).

Since R

3

i

r ——-——dr we find

?J? 2 4- r2

- —-—-——— I(R2 _

2 R r \\

9

--R3

3.17 On measures in shape spaces

v,(E) = 6/TT.

69

(3.17.2)

As for the measure (3.16.7) we have vs(E) = oo.

(3.17.3)

This can be seen from (6.11.1) which gives the values of this measure for a family of sets. An independent confirmation follows from the considerations of §4.6 where vs is viewed as a Haar measure on a certain non-compact group, see (4.6.4). Clearly the measure in the space of tetrahedral shapes which appears as a factor in (3.16.11), namely 2,3,4

is finite. Yet another measure in the same space, namely (3.16.14), is totally infinite. Again, this measure coincides with a Haar measure on a group (see §4.12), and this observation enables us to find explicitly its density in appropriate coordinates. We end with a remark concerning the density sin a sin 2 i//2 sin 2 \j/3 da dij/2 d\j/3

(3.17.4)

which appeared in (3.15.3). It yields a natural geometrical example of three random variables which are not completely independent, yet there is independence for each pair. Indeed, the flat angles \j/l9 \j/2 and \j/3 of the random trihedron described by (normalized) density (3.17.4) have joint densities proportional to sin \j/1 sin \j/2 sin i//3 dij/l d\\i2 dij/3.

(3.17.5)

This result is obtained from (3.17.4) by applying cos \j/1 = —cos i//2 cos \jj3 + sin \\i2 sin \j/3 cos a, a standard formula from spherical trigonometry. Since the variables \l/2 and \j/3 remain intact, we have sin \j/1 di/^ = sin ij/2 sin \j/3 sin a da and it remains to substitute this into (3.17.4). The variables ij/l9 \j/2 and i//3 are not independent since the domain of \j/l9 \j/2 and \j/3 is not a product set. Nevertheless, pairwise independence is there, as proved by integrating out the parameter a in (3.17.4). We end the section with some remarks concerning the spaces £„ of (labeled) shapes of sequences ( ^ , . . . , ^n+1) of points in Un ('practical' cases are n = 2, 3). We will give (in the next section) a detailed description only for £ 2 = £ but a rough idea about the possible construction of labeled tetrahedral shapes was given in §3.16, V and VI, see also §4.12. In the context of point processes (chapter 9) we will consider shapes of sets {0>u..., ^n+1} of points in IRn (of simplexes). For them simplexial shapes

70

3 Measures invariant with respect to Euclidean motions

without labels can be defined and the corresponding space we will denote by En/(n +1)!. This notation reflects the fact that in En we have (n + 1)! labeled shapes which correspond to different labeling of a single shape without labels. The measures on E 2 and E 3 we considered except for (3.16.14) are all symmetrical i.e. invariant with respect to the permutations of numbers attached to the vertices. Any symmetrical measure m on En uniquely determines a measure m* on En/(n + 1)!. One can think that Ew/(n + 1)! is a subset of EB. Then m* can be defined to be the restriction of m on Ln/(rc + 1)!. In particular, m*(En/(n + 1)!) = ((n + l)!)" 1 *!^).

(3.17.6)

3.18 The spherical topology of £ The topological nature of the spaces of shapes of /c-tuples of points in Un has been disclosed by Kendall in [41]; see also [63] and [68] for the presentation in the setting of other work on random shapes carried out in England in recent years. In the special case of triangular shapes, Kendall's result says that, topologically, the corresponding shape space (E in our notation) is the usual sphere S 2 . We show now that this can be deduced from the symmetry and continuity properties of the space E. First we stress that we treat triads as ordered sequences rather then sets. Thus we could call ( ^ , ^ 2 , ^ 3 ) 'labeled' triangles and £ is the space of labeled triangular shapes. We call two triads equivalent if one can be transformed into another by applying a motion from M 2 and then a homothety from the group H of homotheties of the plane (equivalently a scale change in (R2). By definition, the triangular shape a is a class of equivalent triads and E is the factor space where excluded are the triads with totally coinciding points (such triads have no shape), HM 2 is the corresponding group. We denote by p permutations of the indexes (1, 2, 3). We denote by p(0*l9 ^ 2 , ^ 3 ) the accordingly transformed triad (^ l9 &2, ^ 3 ) . For instance, if p sends (1, 2, 3) into (1, 3, 2) then p(Pl9 929 »z) = (&u ^ 3 , 92\

(3.18.1)

If a is the shape of (^ l9 &2, 0>3) then po will denote the shape of p{0>u 0>2, 0>3). Topologically the shapes of the triads from the set U2 x U2 x R 2 \{triads with &x ^ 0>2} = A form the Euclidean plane. This can be seen from the map which is well-defined on the above set: a = ((0, 0), (0, 1), 0>z)

(3.18.2)

3.18 The spherical topology of S

71

(under this map each a e A is identified with certain representative of the class of triads with the same shape a and eventually with ^ 3 e U2). The triads of the set complementary to A, i.e. from the set all have the same shape, a0 say. This a0 can be obtained as the limit of the shapes (3.18.2) under the single requirement that the distance of ^ 3 from the origin tends to oo. Proof We can identify o0 with the triad a0 = ((0, 0), (0, 0), (0, 1)), therefore pa0 e A. In the following p is as in (3.18.1)). We consider the map (3.18.2) as applied to the shapes of po0 and pa, where o is given by (3.18.2). We observe that as ^ 3 goes to oo then lim pa = po0

(in the Euclidean topology of A).

By the properties of the factor topology we can remove p on both sides. Hence the assertion. We see that to obtain the whole L we have to compactify A (the plane) by a single point oQ at infinity. Such a compactification converts (topologically) the plane into a sphere. The latter fact is well known and can be visualized from the usual stereographical projection.

Haar measures on groups of affine transformations

In §1.3 we mentioned the possibility of finding Haar measures on a group, based upon knowledge of Haar measures on its subgroups. In this chapter this method is applied to the groups of Lebesgue measure-preserving affine transformations on Un (n = 2, 3). As a by-product we derive the Haar measures on what we call 'affine deformations groups' and make connections with the position-size-shape factorizations considered in §3.16. These new considerations permit the expansion of the list of explicit factorizations. Their application leads to the solution of the modified J. J. Sylvester problem for four points in U2 and five points in 1R3. This development culminates in the derivation (by means of factorization) of natural probability distributions in the spaces of'affine shapes' of m-point sets in U2.

4.1 The group A® and its subgroups We denote by A ° the group of 2 x 2 matrices with determinant equal to + 1. Equivalently, each A0 e A° is an area-, origin- and 'sense'-preserving affine transformation of U2. ('Sense-preserving' means that for any oriented line g e G its left halfplane is mapped by A0 into the left halfplane of A°g.) Let two orthonormal vectors ex and e2 emerging from the origin 0 be fixed (they determine the x and y coordinates on IR2). Each A° e A° is completely determined by the vectors A°e1 and A°e2 {Aoet is the image of et under A0), see fig. 4.1.1. In fact, the coordinates of A°el and A°e2 when written columnwise constitute the matrix of A0, and the matrix elements (albeit non-independent) can serve for coordinates in A°. However, we will use a different parametrization. Given .4° e A°, its inverse (A0)'1 can be obtained as a product (A0)'1 = CHw,

(4.1.1)

where the transformations C,H,w e A2 are of a very special type. Namely: w

73

4.1 The group A 2 and its subgroups

Figure 4.1.1 The area of the shaded parallelogram equals 1

•

x

y =a

Figure 4.1.2 C maps each horizontal line onto itself. On each horizontal line C acts as a rigid shift. For the line y = a the shift is ac, where c is the shift for the line y — 1

is a rotation by the angle cp shown in fig. 4.1.1; H corresponds to scale changes on the x and y axes, where the homothety along the x axis is chosen from the condition HwA°e1 = el9 while the homothety along the y axis is chosen from the condition that H be area-preserving. Thus H can be represented by the matrix H =

0 10 h~l

h>0;

C is a linear transformation which is explained as follows. Because of the area-preserving property of Hw, the endpoint of HwA°e2 will lie on the horizontal line y = 1. C is a transformation from A £ which leaves the Ox axis intact and maps HwA°e2 on e2. This transformation is shown in fig. 4.1.2.

74

4 Haar measures on groups of affine transformations

Clearly, each C can be represented by a matrix [0

It follows from the caption to fig. 4.1.2 that each C is a Cavalieri transformation of the plane. It is clear from the above that the transformations C, H and w in (4.1.1) are uniquely determined by (A0)'1. By the usual inversion argument we conclude that any A0 e A 2 can be represented as A0 = CHw as well as

(4.1.2) 0

A = w'H'C (representations with different orders of w, H and C elements also exist). Both representations in (4.1.2) are uniquely determined (because of uniqueness in (4.1.1)). The transformations w belong to the group W2 of rotations of U2 (see §3.1). The transformations C constitute a group which we denote by C1. The transformations H also constitute a group, which we denote by H x . The group H x can be identified with the multiplicative group of positive numbers, while Cx can be identified with the additive group of real numbers (i.e. with Tx). On W 2 , Hi and Cx we have the bi-invariant Haar measures dw = dcp (the arc length measure), d/i dH = — h

(the logarithmic measure),

dC = dc

(the Lebesgue measure).

4.2 Affine deformations of U2 The products CH

CeCl9

HeM,

form a group which we denote by V2 and call 'the group of affine deformations of U2\ The elements of V2 are represented by matrices of the form

which we call 'affine deformations' of U2. We note that both representations V = QHr and

(4.2.1)

4.2 Affine deformations of U2

75

V = HXCT

exist and are unique. This follows from the solvability of the matrix equation 0 \/l

fcfVVO

cr\_/l

1/

cx\(hr

0

\ 0 l / \ 0 h~l

which has the solution

CT = Cx' n T

.

(1)

Therefore, in order to find the left d V and the right d(r)V Haar measures on V2 we can use (1.3.6) and (1.3.7), putting X = V2,

U = C 1?

V = Hi.

(1)

Let us find d V both in terms of the coordinates c b hT and cr, hx. We assume that the unknown measures ml and m2 in (1.3.6) have densities, i.e. m^dif) = ptih) dh,

m2(dC) = p2(c) dc.

Equating the right-hand sides in (1.3.6) yields dcx p i i h j dhT = p2(cT)

dcT h x x dhx.

(4.2.3)

Using the solution (4.2.2) we express the measure on the right-hand side of (4.2.3) in the coordinates cx and hT. This yields or Here the variables c, and hT are independent (in the usual analytical sense). Therefore we conclude that p2 = const, p^h) = const -h~3. We choose const = 1 and finally get d^V = K3 dc, dhT = K1 dcr dh{.

(4.2.4)

d ( r ) V = h;1 d c , dhT = hx dcx dhx.

(4.2.5)

Similarly, we find (1)

(r)

We note that our expressions for d V and d V depend on the combination of the subscripts 1 and r. Clearly this is because the two maps

which correspond to (4.2.1) are essentially different and produce on IR x (0, oo) different images of the same measure on V2. Similar situations occur in other sections.

76

4 Haar measures on groups of affine transformations

4.3 The Haar measure on A? From (4.1.2) and (4.2.1) we derive the existence and uniqueness of the representations A0 = V ^ A0 = H\Vr, where A°e/\°2,

w 1? w r GW 2 ,

V,,V r eV 2 .

Therefore the approach of §1.3 can again be used. By Haar factorization, any measure on A 2 which is invariant with respect to the transformations A0 -> WA°w

is necessarily proportional to (see (4.2.4)) d(1)V,dwr.

(4.3.1)

Similarly, invariance with respect to the transformations A0 -• WA°V yields proportionality to the measure d(r)Vr dw,.

(4.3.2)

It is well known (see [2]) that the group A 2 possesses a bi-invariant Haar measure dA°. It follows from our criterion in §1.3 that both measures (4.3.1) and (4.3.2) are proportional to dA°. By the same criterion an independent proof of the bi-invariance of dA° would follow if one shows that the Jacobian relating (4.3.1) to (4.3.2) equals 1 identically. We do this at the end of this section. Anyhow, bi-invariance of dA° being established, we have by (1.3.9) dA° = d(1)V, dwr = d(r)Vr dw,. To obtain explicit expressions for dA°, we use the formula (4.2.4). For instance, if we put V^ffiCa,

i.e.

^ ° = // 1 C 2 w r ,

then we get using natural indexation dA° = h\l dc2 dhx dcp. Putting V, = Q i / 2 ,

i.e.

A0 = Ci/^vvv,

we get dA° = h2

3

dc1 dh2 dq>, etc.

Further expressions can be obtained using (4.2.5). We now outline a calculation of the Jacobian relating the measures (4.3.1) and (4.3.2) which can be a useful prototype for calculations in higher dimensions.

77

Let us represent A0 by a matrix: a-y 0

The equation A = V{ wr in matrix form can be written as \a21

a22)

\0

hi J \sin (px

cos q>x

from which we find

a 12 = — hx sin (jpx + /ix cos cpx • c x . It follows simply from this that da 2 1 dalx da12 = \an\hi1

d/ix dcp^ dc x .

(4.3.3)

Similarly the matrix equation sin cp2

cos % J\ 0

h2c2\ ft2x /

(which corresponds to A0 = w, • Vr) yields da 2 1 daxl da12 = /i||cos
(4.3.4)

The Jacobian in question is J in the relation /i^"1 dhx dq>x dc1 = Jh2 dh2 dq>2 dc2. We gather from (4.3.3) and (4.3.4) that J = 1.

4.4 The Haar measure on A 2 We denote by A 2 the group of all area-preserving affine transformations of IR2; generally speaking they do not preserve the origin 0 e U2. By a direct geometrical argument, resembling that used in §4.1, we can show that each A e A 2 can be represented either as A = txA°T,

with

tx e T 2 , A* e /\°2

A = A?tT,

with

tTeT2,A?e/\°2

or as and both representations are unique. The products denote group multiplication in A 2 (rather than a transformation of a vector by a matrix; for the latter operations we use the sign *). The existence of a bi-invariant Haar measure dA on A 2 is well known (see [2]). Therefore by (1.3.9) dA = dtx dA°T = dtr dA,0,

(4.4.1)

78

4 Haar measures on groups of affine transformations

where dt is the Haar (Lebesgue) measure on T2, and the measure dA° was discussed in the previous section. In fact, bi-invariance of dA (and consequently (4.4.1)) can be established independently without much effort. The solution of the equation

M,° = Aft, is easily seen to be

This transformation is of Cavalieri type, i.e. it preserves any product measure m(dA1°) dtT (because A° by definition preserves the Lebesgue measure drr). Therefore bi-invariance of dA follows by the criterion of §1.3. We get yet another factorization of dA if, for i e A 2 , we use the representations A = MxVn

with

M, G M 2 , Vr e V2

A = V1Mr9 with

M r GM 2 ,V 1 eV 2 .

(4.4.2)

and Since, again, both representations are unique, by the criterion of §1.3 we have dA = dMx d(r)Vr = dMr d(1)V,,

(4.4.3) (r)

(1)

where dM is the Haar measure on M 2 , and the measures d V and d V have been discussed in §4.2.

4.5 Triads of points in U2 A triad (Ql9 Q2, Q3) of non-collinear points from U2 can be represented as (Gi,G2,e 3 ) = ( 4 S , n (4.5.1) where S is the area of the triangle Ql9 Q 2 , Q3, A e A 2 , and 9C is either 1 or - 1 . We determine the parameters A and X as follows. First we choose three non-collinear points ^ , ^>2, ^ 3 e U2 to form a base of our map. In particular we can take ^

= (0, 0), the origin

^ 2 = (1,0) ^3 = (0, 1) By 5 we denote a homothety of U2 corresponding to the matrix S = !

s o

For a given triad (Ql9 Q2, Q3) with S > 0 there are two possibilities:

4.5 Triads of points in R2

79

I. An A e A 2 can be found for which

(61,62, 63) = (the right-hand side stands for the image of (^1,^2,^3) under the product transformation As). In this case we put (61,62, 63) = (A,S,1) II. An A e A2 can be found for which (61,62, 63) = As(Pl9P3, 92) In this case we put (61,62, 63) = (^,S, - 1 ) Since in both cases A is determined uniquely and no other possibilities exist, the construction of (4.5.1) is complete. Remark The 3C = — 1 case corresponds to the situations where a reflection is needed. We could avoid separate considerations of the above two cases if we allowed the value — 1 for det A in §4.1. Clearly (4.5.1) implies that (see §1.6) U2 x R2 x R 2 \{triads of collinear points} « A 2 x (0, 00) x {1, - 1 } . (4.5.2) Any transformation Ax e A 2 acts on the above product in a Cavalieri way, namely (say if 9C = 1) ^1(61, 62, 63) = AtAs(Pl9

^ 2 , ^ 3 ) = (AXA9 S, 1)

Hence, on applying Haar factorization, we get the following proposition. Any measure \i on U2 x U2 x [R2, for which ju({collinear triads}) = 0 and which is A 2-invariant, necessarily has the product representation fi = h/l2x

m,

(4.5.3)

where h&2 is the Haar measure on A 2 , while m i s a measure in the space (0, 00) x { 1 , - 1 } . This result does not depend on the choice of the base triad &>l9 ^ 2 , ^ 3 with which we construct the map (4.5.1). (This follows from the Haar measure properties of h&2.) An obvious example of a measure \i for which the assumptions of the above proposition hold is d 6 i d g 2 d63, where d6, are planar Lebesgue. The corresponding factor measure m can be found from homogeneity considerations (see §1.4). The measure d 6 i d g 2 d63 is of order 3 in area. According to (4.4.1), dA is of order 1 in area. Therefore the corresponding measure m should be of order 2 in area. Thus, necessarily,

80

4 Haar measures on groups of affine transformations

m(dS x X) = c^S dS,

X = 1, - 1 .

It follows from reflection invariance of d<2x dQ2 dQ3 that cx = c_x = c. We show at the end of the next section that c = 4. Thus, dQi dQ 2 d<23 = 4dAS

dS.

(4.5.4)

As another example we consider the measure \i in the space (4.5.2) defined to be the image of the (bi-invariant, see [2]) Haar measure on the affine group Af (i.e. the group of 2 x 2 matrices with non-zero determinants) induced by the map (<2i, G 2 , Q3) = **&!, &i> &z\ ^ € AJ. By a direct application of the principles of §1.3 we find that the Haar measure on Af has the form dA* = cS,'1 dS, dAT = cS;1 dSr dA{ where s b Ar and 5r, Ax are defined by the relations (or their versions which include reflections). Thus using natural indexation MdGi dQ2 dQ3) = cS,-1 dS, dAT = cS^1 dSr ( U , The constant c > 0 is present because the measure ft was defined up to a constant factor.

4.6 Another representation of d(r)V The result (4.5.4) is closely related to the factorization of dQx dQ2 dQ3 given by (3.16.8). Indeed, from (4.4.3) we find that dQi dQ2 dQ3 = 4 dMj d(r)Vr S dS

(4.6.1)

<*Gi d g 2 dQ 3 = 4 dM r d(1)V, S dS.

(4.6.2)

and We stress that the 'variables' Ml5 Vr and M r , V! above are defined by the relations (Gi? 62, G3) = M and (Gi, G2? G3) = Vi or their versions that include reflections. Here ^ l 5 ^ 2 » ^3 form the base of the maps (compare with §4.5). Let us compare this with (3.16.8). A natural question arises as to whether we can eliminate the measures dM S dS and thus derive a new interpretation for the measure vs on £. The

4.6 Another representation of d(r)V

81

answer is that we can do this with (4.6.1) but not with (4.6.2). The reason is that, for any M G M 2 , while the parameters M r and Vj lack similar Cavalieri properties. The elimination of the dMx S dS factor yields (see (4.2.5)) vs(da) = 2 d (r) V = 2k;1 dcx dhT = 2hx dcT dhx.

(4.6.3) (r)

It follows from the considerations of §4.5 that vs is the image of d V under the maps V2 x {1, - l } - > £ ,

(4.6.4)

each such map depending on the choice of the 'base' triangle &>l9 ^ 2 , ^ 3 . Thus, as soon as the latter triangle is fixed, we put (V, 1) -> shape of the triangle V ^ , V ^ 2 , ' and (V, - 1 ) -• shape of a reflection of V^ x , V^ 2 , The independence of the image of d(r)V from the choice of the 'base' triangle clearly follows from the Haar measure properties of d(r)V. Let us check (4.6.3) by a direct calculation. We choose ^

= (0, 0),

92 = (1, 0),

^ 3 = (0, 1)

for our 'base' triangle. From fig. 4.6.1 cot ^ = cx cot £ 2 = (hr - cx/hT)hr = h2x - cx. Hence, -sin2^

0 —2hT sin2

sin" = 2hT sin 2 (^ sin 2 £ 2 dhT dcx. Using (3.16.7) we get

Figure 4.6.1

dhT

dcx

82

4 Haar measures on groups of affine transformations

dvs = (sin £x sin £2 s i n ^ + £2))~1 &€i d£i _ r sin gt sin g2 d ^ d ^ fir d/i r dc, COt ^ + COt £ 2

= 2/171 d q dfcr, i.e. (4.6.3).

4.7 Quadruples of points in U2 We consider the space of sequences (Ql9 Q2, Q3, Q4) of points in the plane with no three points on a line. This space is essentially U2 x U2 x U2 x R 2 \Z, where Z = {quadruples possessing collinear triples}. Using the representation (4.5.1) we can write (61,62,63,6*) = ( 4 $,«•,&), where A, S and 9£ refer to the triangle 616263Let fi be any measure on U2 x [R2 x U2 x [R2 which is A2-invariant and fi(Z) = 0. In order to apply Haar factorization to pi (i.e. to separate h&2) we need to determine Q4 by means of A2-invariant parameters. Let us consider the case shown in fig. 4.7.1. In this case the values Sx = area of the triangle 626364 and S2 = area of the triangle 616463 determine the position of 64, and we have the Cavalieri property 62, fi3,64) = (AXA, s, ar, sl9 s2). Hence, on the set defined byfig.4.7.1, AX(Q19

Figure 4.7.1 The triangle 616263 encloses Q4

4.7 Quadruples of points in U2

83

Figure 4.7.2 hx and h2 are perpendicular to the sides /x and l2

<\h

Figure 4.7.3 The angles a here and in fig. 4.7.2 coincide

(4.7.1)

H = fcA2 x m, where m is some measure in the space of sequences (5, #*, S l9 S2). Let us find m in the decomposition (4.7.1) for the case

where each d 6 t is a planar Lebesgue measure. In the change of variables that follows, A, S and 9£ (equivalently, 616263) remain intact. Therefore we can treat them as fixed. We have see fig. 4.7.2 for notation. The area element d 6 4 which corresponds to dh1 and dh2 is shown in fig. 4.7.3. We have d 6 4 = (sin a)" 1 dhx dh2 = = 2-

S,dS2 \x\2 sin a

i dS2

(4.7.2)

Together with (4.5.4) this yields d 6 i d 6 2 d 6 3 d 6 4 = 8 dA dS dSx dS2.

(4.7.3)

We stress that here S is the area of the triangle 616263 (see fig. 4.7.1). Let us

84

4 Haar measures on groups of affine transformations

Figure 4.7.4 St is the area of 8 1 8 3 8 4 and S2 is the area of 8 1 8 2 6 4

consider the cases complementary to that shown in fig. 4.7.1. Continuation of the sides of the triangle QxQ2Q3 separates the plane, as shown in fig. 4.7.4. The cases in which <24 lies in the angular (shaded) domains can be reduced to the above cases by a change of numeration of points. If the points Ql9 Q2, Q3 and Q4 form a convex quadrangle, then the position of Q4 can be determined by the values Sx and S 2 , the areas of the triangles shown in fig. 4.7.4. A calculation in the style of (4.7.2) ensures that in this notation (4.7.2) remains valid, with S still denoting the area of QxQ2Q3.

4.8 The modified Sylvester problem: four points in U2 Let us denote by V the area of the minimal convex hull of the points Ql9 Q2, Q3 and Q 4 . In the case shown in fig. 4.7.1 V = S. Therefore, when we replace 5, Sx and S2 by the new coordinates

y. -4-

"4

in (4.7.3) we obtain d<2i d g 2 dQ3 dQ 4 = 8 dA V2 dV du dv.

(4.8.1)

It is clear that the range of u and v is the triangle shown in fig. 4.8.1. In the case of fig. 4.7.4 we first replace the coordinates 5, S1 and S2 by

V = S1+S2

s =s (the Jacobian is 1), and then V =V u = S/V v = SJV

4.8 The modified Sylvester problem: four points in 1R2

O

85

1

Figure 4.8.1 The area of the triangle is \ (the Jacobian is V2). In these coordinates we again obtain (4.8.1). However the range of variables u, v is different from that in the case of fig. 4.7.1. Namely the range of w, v corresponding to fig. 4.7.4 is the unit square. Summarizing, we find that the complete range of the w, v parameters is a union of seven components, of which four are triangles identical to that in fig. 4.8.1 and the remaining three are unit squares. From a geometrical point of view, a pair w, v defines what we call the affine (or A 2 reflection and homothetyinvariant) shape of the quadruple Ql9 Q2, Q 3 , <24. We denote an affine shape by T, thus T

= (w, v)

In §4.16 we give a complete topological description of the space x2>4 of affine shapes. If the cases where we have collinearities are discarded then what remains of x2 4 can be represented by the range of the u and v parameters essentially as described above. The reason for the indexes 2 and 4 is that later on (in §4.15) we consider the spaces xnm of affine shapes of m-tuples of points in W (thus here n = 2, m = 4). We find from the above crude description of x2A that the total dw dvmeasure of this space equals 5. Therefore we define a probability measure P 2 4 on T 2 4 putting The main result of this section can now be written as d d dQ 2 d g 3 dQ 4 = 40 dA V2 dV P 2 , 4 (di).

(4.8.2)

The probability distribution P 2 ? 4 is of special interest because it reappears in the context of planar Poisson point processes (§9.7). However, it is of interest in its own right that certain events in x2A have very clear geometrical interpretation and their P 2 ^-probabilities can now be found without difficulty. Remarkably this is the case for the events Bx and B2 originally considered by J. J. Sylvester (recall his famous Vierpunktproblem; see [35] and [55]), where

86

4 Haar measures on groups of affine transformations

Bl9 the minimal convex hull of 21226364' *s a triangle; B2, the minimal convex hull of 61626364? *s a quadrangle. In the J. J. Sylvester formulation the four points were dropped independently, each with uniform distribution in a convex domain D c R 2 ; hence, the probabilities of Bx and B2 depend on D. In the present modified version we ask for the values of ^ , 4 o n #i a n d #2- From the above crude description of x2,4 it is clear that B1 = the union of the four triangular components, B2 = the union of the three squares. Thus the solution of the modified Sylvester problem is

4.9 The group A3 and its subgroups We denote by A3 the group of 3 x 3 matrices with determinants equal to + 1 . Equivalently, A3 is the group of volume, origin- and 'sense'-preserving affine transformations of U3. Let three orthonormal vectors el9e2 and e3 emerging from O be fixed (they determine the x, y, z coordinates in U3). Given A0 e A3, its inverse (A0)'1 can be represented as a product (A0)'1 = HCw,

(4.9.1)

where the transformations H, C, w e A3 now depend on more than one parameter each. They are uniquely determined as follows: w is a rotation of 1R3 (it depends on three parameters, see §3.2). In (4.9.1) w is determined by the condition that wA°ei has the direction of el9 wA°e2 lies in the plane ele2 (i.e. z = 0), and the y coordinate of wA°e2 is positive. The transformation C in (4.9.1) is a product of two Cavalieri transformations from A3: C = C (2) C (1) . C

(1)

belongs to the group C C (1)

(1)

(4.9.2)

of matrices:

1 a 0 = 0 1 0 \0 0 1

— 00 < a < 00.

(4.9.3)

Under C(1) the plane y = y0 is shifted by the vector (yoa, 0, 0) (in particular, the x axis remains intact), —00 < y0 < 00. We choose C (1) from the condition C(1)wA°e2 has the direction of e2 (note that C (1) is isomorphic to the group C of §4.1 and acts similarly on the z = 0 plane).

4.10 The group of affine deformations of R3

87

C(2) belongs to the group C(2) of matrices: /I 0 b\ (2) -oo < b < oo, -oo < c < oo. (4.9.4) C = 0 1 c\ \0 0 1/ Under C(2) the plane z = z0 is shifted by the vector zo(b, c, 0) (in particular the xOy plane remains intact), — oo < z0 < oo. We choose C(2) from the condition C(2)C(1)wA°e3 has the direction ofe3. It follows that the product (4.9.2) sends each wA°eh i = 1,2,3, onto the positive semiaxis determined by the corresponding e{. In (4.9.1) H belongs to the group 0-02 of matrices: In 0 0 \ if= 0 v 0 « > 0 , i;>0. (4.9.5) 1 \0 0 (uv)- ! Under H the scales along the x and y axes change independently by the factors u and v, and we have a compensating scale change along the z axis. We find H from the conditions HCwA°el = e1 (this determines u), HCwA°e2 = e2 (this determines v). All transformations considered were volume-preserving. Therefore, necessarily, HCwA°e3 = e39 and the construction of (4.9.1) is now complete. From (4.9.1) we immediately conclude the existence of (unique) representations valid for every A0 E /\^: A0 = HCw (4.9.6)} A0 = WCH' (representations with different orders of elements H, C and w also exist). The subgroups we consider possess (unique) bi-invariant Haar measures. The Haar measure on C(1) is da (the linear Lebesgue measure); on C(2) it is db dc (the planar Lebesgue measure); and on H2 it is (MI;)"1 du dv (the product logarithmic measure). The Haar measure on W3 was considered in §3.3.

4.10 The group of affine deformations of U3 First we note that the products C (2) C (1)

C (2)

g

C(2)9

C (l)

e

C (l)

constitute a group which we denote by C2. Equivalently, an element of C2 can be represented by a matrix of the type

4 Haar measures on groups of affine transformations

'1 0 \0

a b\ 1 c\ 0 1/

-oo
(4.10.1)

Let us consider the Haar measure on C 2 . It is easy to check that the matrix equation

C^C™ = a2)CiX)

(4.10.2)

is equivalent to the following equation system involving the parameters a, b and c (see (4.9.3) and (4.9.4)); the lower index will refer to the corresponding matrix in (4.10.2): ar = ax cr = c x bT = bx + axcx. This is a Cavalieri-type transformation, hence da! dbT dcT = dar dbx dcx.

By the criterion of §1.3 we conclude that the group C 2 possesses a bi-invariant Haar measure. In terms of the elements of the matrix (4.10.1), this measure is da db dc. We call the products V = HC,

HeH2,

CeC2

3

affine deformations of [R . They constitute a group which we denote by V3 (the group of affine deformations of R3). It is important that both the representations V = HxCr and

(4.10.3) V = CXHT

exist and are unique. This enables us to apply the considerations of §1.3. The matrix equation CxHT = HxCr,

Cx,CTeC2,

Hx,HreH2

is equivalent to the following equation system, where the elements of the matrices (4.9.5) and (4.10.1) have indexes which refer to their order in (4.10.3): uT = u x vT = vx axvT = uxaT

(4.10.4)

Equating the right-hand side expressions in (1.3.7) (under the assumption of existence of densities px and p 2 of mi and m2)

4.11 Haar measures on A3 and A 3 Pi( a i5 ^i> ci) dax dbx dciiUfVr)'1

duT dvT = p2(ux,

vx) dux dvx daT dbT

89 dcr.

Changing to the variable ul9 vu aT9 br, cr on the left-hand side by using (4.10.4) yields ufv^icMVi2, aTuxvx~\ bTufvx) = p2(ux, vx\ i.e. pi = const,

P2(ux, vx) = ufvx.

Thus we have found two expressions for the right-invariant Haar measure d ( r ) VonV 3 : dV = ( i i ^ r 1 dur dvT dax dbx dcx and

(4.10.5) dV = ufvx dux dvx daT dbT d c r .

4.11 Haar measures on A3 and A 3 The first step towards the derivation of the Haar measure on A 3 (this measure is bi-invariant, see [2]) can be taken by using the representations A0 = V,w J Ao _

wl5 w r GW 3 ,

V,,V r eV 3

since their existence (and uniqueness) follow from (4.9.6). Let us denote by dA° the Haar measure on the group A 3 . By the necessity part of the criterion in §1.3 we find that cL4° = d(1)V! dwr = d(r)Vr dw,,

(4.11.1)

where dw is the Haar measure on W 3 . We also consider the group A 3 of area- and 'sense'- (but no longer origin-) preserving affine transformations of R3. For every A e A 3 we can easily derive the following unique representations: A= tl T

MreT3>

_*

A^A°re/\%.

The existence of bi-invariant Haar measure on A 3 is well known [2]. We denote this measure by dA. By the criterion of §1.3 we conclude that dA = dtx dv4r° = dtT dAx°.

(4.11.2)

In the next section we will use the representation dA = dMx d(r)Vr = dM r d(1)V,

(4.11.3)

which follows from the existence and uniqueness of representations, A = MXVT A = WXMT,

Mx, Mr G IU3, Vr, V! G V3

Here dM is the Haar measure on the Euclidean group M3 (compare with (4.4.3)).

90

4 Haar measures on groups of affine transformations

4.12 V3-invariant measure in the space of tetrahedral shapes By means of a map 'based' on an appropriately fixed s e q u e n c e ^ , ^2> ^3> &**) of points in [R3, any sequence (Ql9 Q2, Q 3 , Q4) of non-coplanar points in IR3 can be represented as

(Qi,Q2,Q3,QA) = (A>v9n where A e A 3 , J = 1 or - 1 , and V is the volume of the tetrahedron with vertices Ql9 Q2, g 3 , Q4 (compare with §4.5). The variable A is determined from the relation (Qi, Qi, G3> 64) = Av(Pl9 ^ 2 , ^ 3 , ^ 4 ) , case ar = 1 (or its version which includes a reflection when 9C = — 1). Here ins a homothety of IR3 which corresponds to the matrix

l*/v 0 = 0 \ 0

0

0

*/v 0 0 */V

The measure PJf=1 dQt (where each dQ( is Lebesgue in IR3) is invariant with respect to A 3 . Therefore, by homothety considerations, n dQi = cdAV2dV

= c dM, d(r)Vr V2 dV,

(4.12.1)

i=l

below we calculate the constant c to be c = 3-6 3 .

(4.12.2)

We can equate the right-hand sides of (4.12.1) and (3.16.14). Applying elimination of measures we find mv(d(j) = 3 • 6 3 d(r)V.

(4.12.3)

This reveals certain invariance properties of the measure mv (compare with the discussion in §4.6). A direct derivation of (4.12.3) can be of interest. It is convenient to take a general affine deformation in the form l 1 a b ju 0 0 ju av b{uv)~1 \ V ciuv)' V= 0 1 |-0 V 0 1 ) 0 (uv)! 0 (uv)-1 0 0 1/ o o We take the vertices of the 'basic' tetrahedron to be

=°

^

= (0, 0, 0),

^ 2 = (1, 0, 0),

^ 3 = (0, 1, 0),

^ = (0, 0, 1)

and therefore the coordinates of the points V^-, i = 2, 3, 4, can be read from the matrix representing our V. This enables us to easily express the shape parameters of the tetrahedron which we used in §3.16, VI, in terms of a, b, c, u and v. We have (seefig.3.16.4)

4.13 Quintuples of points in (R3 COt V3 =

91

a,

COt V4 = b/y/(l + C2\

cot O 4 = c, as well as a = vu~\\ + a 2 )" 1 , hence the problem is reduced to two- and three-dimensional Jacobian calculations. We easily find that dv3 dv4 dO 4 = [(1 + a2y\\

+ b2 + c 2 r V V ( l + c 2 ] da dfe dc

and da dp = 3(ii4i7)-V(l + a 2 )V(l + b2 + c2) dw di>. Substitution of these expressions, together with (sin v 3 )" 2 = 1 + a2, (sin v4)~2 = (1 + b 2 + c 2 )(l + c2)~\ (sin ^ 4 ) " 3 = (1 + c 2 ) 3/2 into (3.16.13) and (3.16.14) yields f ] dQt = 3-63dMV2

dV{uv)~l da db dc du dv.

i= l

Because of (4.10.5) this is exactly the same as (4.12.3).

4.13 Quintuples of points in R3 We consider the sequences Ql9 Q2, Q 3 , Q 4 , Q5 of points in U3 which do not possess coplanar quadruples. The purpose is to factorize the product of Lebesgue measures

ride, £=1

into a product of dA (Haar measure on A 3 ) and a measure in a space of A 3 -invariant parameters. We will denote by Vt the volume of the tetrahedron {Qi}t=i\Qi (thus the volume of Q^QsQ* is V5). We will use (4.12.1) (with V5 replacing V) in conjunction with the expression of dQ5 in terms of the volumes Vl9 V2 and V3, say. The continuation of the faces of Ql9 Q2, Q 3 , 2 4 splits the domain of Q5 into the following connected components: (a) the interior of QiQ2Q3Q4, (b) the four infinite trihedral regions,

92

4 Haar measures on groups of affine transformations

(c) four infinite quadrihedral regions, each having a face of 61626364 its boundary. (d) six infinite quadrihedral regions, each having an edge of 61626364 the boundary.

on

on

Within each of the above regions we have d6s = edVld^dV\

(4.13.1)

thus always f ] d6, = 3 • 6 4 dA dV5 dVx dV2 dV3.

(4.13.2)

The proof of (4.13.1) when Q5 belongs to (a) can be as follows. Denote by Si - the area of the triangle {61} 1 \6f • ^ - the height of 65 above St. Then for i = 1, 2, 3 d^ = ^S( dht

and for a parallelepipedal d6s

n 3dK 1,2,3

St

An elementary expression for the constant ft (which depends on the trihedral angle at 64) is of no significance in this calculation. The same formula yields

n -a 1,2,3 therefore

and we get (4.13.1) by substitution into (4.13.3).

4.14 Affine shapes of quintuples in U3 Following the ideas of §4.8 we rewrite here the product dF 5 dFx dK2 dF 3 (see (4.13.2)) in terms of the volume V of the minimal convex hull of Ql,Q2,Q2), Q4., 65 and of the ratios VJV, where the latter define affine (A3-invariant) shape of the quintuple. The formulae used for changing to new variables are different for different regions of variation of 65. By symmetry, it is enough for our purposes to consider component (a), one component of (c), and one of (d) (see §4.13) (see Table 4.14.1).

4.14 Affine shapes of quintuples in R3

93

Table 4.14.1 Region of Q 5

Domain of T^

Intermediate variables

positive)

Volume of the minimal convex hull of Q l 5 <22> g 3 , 64> Qs

(1) Type (a)

Vl + V2J -V3
V=V5

(2) Type(c) The face is 616263

V, + V2^-V3>V5

V=V1 + V2- ^ 3

v2,v3,V vl + v2+ V3
V2, V3, V5 (all

(3) Type(d) The edge is 6 3 6 *

Domain of the intermediate variables (all positive)

v, + v2
V3
V=Vl + V2-

v2,Vz, V v*
In each of cases (1), (2) and (3), the change from Vu V2, V3, V5 to the intermediate variables (in the fourth column) is with Jacobian J = 1. Now we introduce the affine shape variables T = (T1,T2,T3).

Namely, in cases (1) and (3) we put while in case (2) T1

=

From these remarks it follows that

fl

= 3 - 6 4 d i F 3 dFdii dt 2 di 3 ,

(4.14.1)

The range of T can be seen from the last column of table 4.14.1 to be case (1): TX > 0, T 2 > 0, T 3 > 0, TX + T 2 4- T 3 < 1 (a simplex of volume £); cases (2), (3): i{ > 0, i = 1, 2, 3, T1 + T 2 < 1, T 3 < 1 (a prism). Consequently, the space T 3 5 of affine shapes that we consider is a union of five simplices and ten prisms as above (see the regions described in §4.13). Hence the dx1 dr 2 dr 3 volume of x 3 5 equals ^ . We introduce a probability measure on x3 5 : which allows (4.14.1) to be rewritten as follows: n dQt = 105-6 3 dA V3 dV P 3 , 5 (di).

(4.14.2)

94

4 Haar measures on groups of affine transformations

This result is similar to (4.8.2); generalization of both equations is given in the next section. With the same motivation as in §4.8, we can consider a version of J. J. Sylvester's problem for the probability distribution Q \ 5 . Namely, we wish to find the probabilities of the events £3, the minimal convex hull of Qi62Q364G5> i s a tetrahedron, and of its complement £ 4 , the points 2 i 6263 6465* n e o n t n e boundary of their minimal convex hull. Clearly, event B3 corresponds to the union of the regions labeled (a) and (b) in §4.13, or, more properly, to the union of thefivesimplex components of x3 5 . Therefore and

4.15 A general theorem Equations (4.8.2) and (4.14.2) are special cases of a general result [20] concerning the spaces xnm of m-point affine shapes in Un and certain probabilities Pnm on these spaces xntVn (see §4.8) For simplicity we formulate and prove this theorem for the planar case. Most of the notation used here is the same as before. We consider sequences (61 J • • -5 Qm\ Qi e ^ 2 where Ql9 Q2, Q3 are non-collinear. From each class of A 2-equivalent sequences we choose an element of the form where ^

= O (the origin)

^2 = (1, 0) ^ 3 = (0, y) with some y =£ 0. We put # • = 1 ify > 0 -lify<0. We denote by V the area of the minimal convex hull of the set {^, i^2> ^3> 64, • • •, 2m} ( t m s value is constant for all A2-equivalent sequences). In the case f = 1 we denote by T the image of the sequence ((0, y), Q g m ) after a homothety of U2 which reduces the area of the minimal convex hull of {&u »2, ^ 3 , Ql,..., Q°m} to 1. To obtain x when X = - 1 we also make a reflection with respect to the x axis.

4.15 A general theorem

95

The equivalence classes in question consist of elements

P29P39Ql...9Q°l),

Ae/\2

(A acts on a sequence). Taken together with the above this defines a map (Qi,...,Qm)

=

(A9ar9v9T).

It has the properties that for any Ax e A 2 A1iQ1,...,Qm)

= (A1A9X9V9T)

(4.15.1)

and (n °f v

T\

— (&p &p op n°

n°\

ij, U, v, T) — ytrl9 tr2, tr3, {J4, . . . , \Jm)

where 11 is the unit element of the group A 2 . The (2m — 6)-dimensional parameter T thus defined is invariant with respect to A 2 , reflections and homotheties. We call T the affine shape of (Q 1 ? ..., Qm). We now consider the product of planar Lebesgue measures dQt. For every m = 3, 4 , . . . ft dQt = cm AA Vm~2 dFP 2 , m (di)

(4.15.2)

i

where cm is a constant, dA is the Haar measure on A 2 and P 2 m is a probability measure in the space t 2 m. Proof The separation of the measures dA, Vm~2 dV and a measure m on the space xlm follows from general principles of Haar factorization and homothety considerations (see §§1.3 and 1.6). Thus it remains to prove that the measure m is totally finite. Let Kr be the disc of radius r centered at O. We have

(4nr= f •• f f[dQt JK2

= 2 Jx2,w

JK2 1

dA Vm~2 dV

m(di)

(4.15.3)

JJBZ

where Br = {(A, V): the set corresponding to A, V, T lies within X 2 }, and we assume that 3C = 1. Let (Ql9..., Qm) be an arbitrary sequence for which V ^ 1 and whose aifine shape is T. We can assume that Qx and Q2 lie at maximal distance, i.e. \Qi,Q2\>\Qt,Qj\ for aU 1,7. It is elementary to show that, by choosing appropriate transformations M e M2 and H e H ^ w e can reach the effect that both the points HMQ1 and HMQ2 lie on the x axis and have the abscissae — 1 and 1 while all other Qf = HMQi lie within the square ( - 1 , 1) x ( - 1 , 1). This is illustrated by fig. 4.15.1.

96

4 Haar measures on groups of affine transformations y

Qt Figure 4.15.1 Q* and g j

are

the points which have maximal (among Qf) distance from the x axis

The minimal convex hull of {Qf} covers the interior of 6*62636*; hence hi + h2 ^ V < 1,

i.e. /*! ^ 1, /z2 ^ 1.

We now denote by X(F, T) an element of A 2 which transforms (U, 1, V, T) in the way described in fig. 4.15.1. We also put Clearly because of (4.15.1), for every T,

n

dA Vm

dV ^

Bx

where

f1 F m , d F fI JO

OLA(V,T) =

dA,

JOC4(F,T)

{AA(V,T):AE(X}.

Because dA is bi-invariant Haar, the inner integral does not depend on F, T and equals ^A2(a) > 0. We conclude that dA F m " 2 d F ^ fcAa(a). Now it follows from (4.15.3) that (4n)m > 2hA2(QL)m(x2tm),

i.e. the measure in question is totally finite. This icsult can be generalized to points in R3 and higher dimensional spaces as suggested by the results of §4.14. Values of P 2 5 for certain events in T 2 5 have been found in [21].

4.16 The elliptical plane as a space of affine shapes In preceding sections we considered slitted versions of the spaces xnm. Now our aim will be to give the topological description of the space t 2 4 . Similar results for general xnm are seemingly unknown.

4.16 The elliptical plane as a space of affine shapes

97

The general course of our reasoning resembles that of §3.18, i.e. we use the symmetry and continuity properties of the space T2,4 which follow from its definition as a factor space. First we stress that we consider ordered quadruples ( ^ , ^ , ^ 3 , ^ 4 ) of points in R2 (rather than sets). Also here we prefer to consider the affine group A J of 2 x 2 matrices with non-zero determinants. Two quadruples are declared equivalent if one can be transformed into another by an affine action. An affine shape T can be formally defined to be a class of equivalent quadruples. The space x2A is defined to be the factor space l(U2 x U2 x U2 x R 2 )\{the set of totally collinear quadruples}]/Af. By definition, totally collinear quadruples do not possess any affine shape. Let us denote by Es the set of affine shapes of the quadruples from (U2 x U2 x U2 x [R2)\{quadruples with &i90>j90>k collinear}, where /,; and k are all distinct and assume values from {1, 2, 3, 4} and s ^ U j or k. Each Es is homeomorphic to the Euclidean plane as seen from the following representations (maps) which are defined on different sets Es: on E 4 ,

T

= ((0, 0), (1, 0), (0, 1), 0>4);

where T e Es is identified with a certain element chosen from the corresponding equivalence class and eventually with 0>s e U2. In this sense we will write Clearly the same affine shape can be represented by points on different planes Es. One can also use representations of affine shapes on affine transformed planes Es. Let x s , ys be the coordinates of 0>s on the plane Es (see fig. 4.16.1, where we show £ 4 and E3).

Figure 4.16.1 The points ^ 4 and ^ 3 on the left and right diagrams roughly correspond to the same affine shape

98

4 Haar measures on groups of affine transformations

The relation between the affine shape coordinates (x 4 , y 4 ) and (x 3 , y3) is as follows: x3 =

—x -,

1 y3 = —

whenever y 4 # 0.

(4.16.1)

P r o o / First assume that y4 > 0. Then to bring the points (0, 0), (1, 0), ^ 4 , respectively, onto (0, 0), (1, 0), (0, 1), it is enough to apply a transformation C eC1 (see §4.1) with c = — x4/>>4 and then to rescale the y axis by the factor l/)>4. Hence (4.16.1). If y 4 < 0 then we first make a reflection with respect to the x axis: (x 4 , yA) goes to (x 4 , |y 4 |) and (0, 1) into (0, - 1 ) . Then we apply the same transformation as above. Because under C e Cx the points (0, - 1 ) and (0, 1) shift in the opposite directions to equal distances, (0, —1) will go to (x 4 /|y 4 |, — l / l y j ) which again corresponds to (4.16.1). Let us consider the following subsets of x 2 , 4 : B2, the affine shapes which on the plane EA lie on the x axis; £ 3 , consists of one point (affine shape) which on both planes £ x and E2 is represented by (0, 0). The sets Bt are pair wise disjoint and B1 u B2 u B3 = x2A. Proof If for a quadruple ^ = 0>2 t n e n ^i» ^3> ^ 4 a r e necessarily noncollinear (otherwise ^ , 0>2, ^ 3 , ^*4 would be totally collinear but this possibility was excluded from the start). Hence, by an affine transformation, ^ , ^ 3 , ^ 4 can be mapped onto (0, 0), (1, 0), (0, 1). Therefore the corresponding T belongs to E2, and moreover it will be represented here by (0, 0). In other words, 0>x = £P2 implies that T e B3. If ^ T* ^ 2 t n e n w e n a v e t w o mutually excluding subcases: (a) ^ 1 , ^ 2 > ^ 4 a r e non-collinear; then T e £ 3 = 5 1 ? and (b) ^ , ^2> ^ 4 a r e collinear; then ^ , ^2, £?3 are necessarily non-collinear. After mapping ^ , ^ 2 , ^ 3 on (0, 0), (1, 0), (0, 1), ^ 4 will fall on the x axis, i.e. T e B2. Our main conclusion will be based on the following facts. (1) In the topology of x2 4 the set B2uB3 is homeomorphic to a circle. Equivalently, as |x| -• 00 the affine shape (x, 0) e £ 4 converges to (0, 0) e E1. Proof Denote by Ao the reflection of U2 with respect to the line y = x. Suppose x > 0. By rescaling the x axis by the factor 1/x we map (x, 0) e E4 into (1/x, 0) G A0E2. But a sequence which converges to (0, 0) in A0E2 also converges in E2 to the same point. The case x < 0 can be reduced to the above by a reflection.

4.16 The elliptical plane as a space of affine shapes

99

(2) Any x e B2 U £ 3 corresponds to a bundle of parallel lines in E3 in the sense that T is a limit (in x2 4 ) of a point moving away in E3 along any line of the bundle, in any of the two possible directions on the line. Different points of the circle B2 u B3 correspond to different bundles. Proof Let us consider the case x e B2 so that x = (x, 0) e £ 4 . The corresponding bundle of parallels on E3 happens to be the image of the bundle of lines through (x, 0) on £ 4 . This follows from (4.16.1) after substitution of the equations of the latter bundle, x 4 = x + / cos i//,

yA = I sin ij/,

to obtain x 3 = — xy3 — cot \jj.

We conclude that the direction of the line in the bundle is determined solely by x, and the horizontal shift of a line in a bundle is cot ij/. The reader may check that the point x e B3 corresponds to the bundle of horizontal lines on £ 4 . Properties (1) and (2) imply that t 2 ) 4 is obtained from E3 in the same way that the projective (elliptical) plane is constructed by complementing a Euclidean plane with a line at infinity' (B2 u B3 in our case). We conclude that, topologically, T 2 4 = E 2 .

Combinatorial integral geometry

In this chapter we consider in detail two topics from the vast new field mentioned in the chapter title, namely combinatorics of lines on U2 and of planes in IR3. The theory has important applications in geometrical processes which we demonstrate in chapter 10 for the planar case. The combinatorics of planes is applied in this chapter in the construction of what we call flag representations of bounded centrally-symmetrical convex bodies in IR3. The last section contains a brief synopsis of other ramifications of combinatorial integral geometry (a complete presentation can be found in [3]).

5.1 Radon rings in G and G In this section we formulate the results of the theory referring to the M2invariant measures in the spaces G and G (see §§2.1,2.2 and 3.6). Two different proofs are given in §5.2 and §5.3. We denote by m, in the following a finite set of points on U2: Given m, let us call two lines gu g2 e G equivalent if both have the same subset of m, in their left halfplanes (see fig. 5.1.1). The bundles of lines through different points of m decompose G into subsets of equivalent lines which we call 'atoms'. There are only two unbounded atoms. Their lines leave the whole of m in one halfplane, which is left for one atom and right for another. The Radon ring T(m) is defined to be the ring generated by all bounded atoms. Any B e 7(m) is a union of bounded atoms. The term 'Radon ring' was introduced in this context in [3]. We stress that the atoms are pairwise disjoint. Let gtj be the line through &>i9 0>j e m directed from ^ to ^ . If in the set m no three points lie on a line, then each gi} belongs to the boundary of exactly

5.1 Radon rings in G and G

101

Figure 5.1.1 gx and g2 are equivalent

Figure 5.1.2 The sign + or — over an index refers to the right or left position of the corresponding point

four# atoms. We can obtain lines from these four atoms by subjecting g{j to sufficiently small displacements, as shown in fig. 5.1.2. It is convenient to consider the displacements as infinitesimal. We have the following proposition. For every set B s7(<m) its M 2 -invariant measure fi(B) can be calculated as a linear combination of the distances pu between points &i9 ^ e m with integer coefficients: fi(B) =

(5.1.1) (ij)

The integers c^B) can be calculated by means of the 'four indicator formula':

cu(B) = IB(l j) + IB(l j) - IB(l )) - IB(l

~j\

(5.1.2)

where IB( i9 j) are the values of the indicator IB(g) on the lines (or, rather, on corresponding atoms) shown in fig. 5.1.2. Let us now define similar rings in the space G. Given a set m, two lines gl9 g2eG are called equivalent if they induce the same separation of m. Bundles of lines through each ^ e <m split G into sets of equivalent lines, which we call atoms. There is only one unbounded atom, namely the one whose lines leave the whole of m in a halfplane. By definition, r{m) is the minimal ring containing all bounded atoms. Every B e r(m) is a union of a finite number of atoms.

102

5 Combinatorial integral geometry

Alternatively, r(m) can be defined as the image oiT{m) under the directionerasing map G -» G. By considering this map it is possible to deduce from (5.1.1) the corresponding decomposition for r(<m). We formulate the following result. Let ^ b e a set with no three points collinear, and let /i be the fVO2-invariant measure on G. For every B e r(m) we have lx{B) = X cu{B)pu,

(5.1.3)

where c{j(B) are integer coefficients. For the calculation of the values of ctj{B) we can again apply (5.1.2), since IB is also naturally defined on directed lines. Since in (5.1.2) now always Cij(B)

=

Cji(B)

the way we ascribe a direction to the line through the points 0>{ and ^ is irrelevant. Examples of calculation by means of (5.1.3) are given in §5.4.

5.2 Extension of Crofton's theorem Remarkably, the result of the previous section follows rather directly from Crofton's theorem, previously mentioned in §1.8. The key fact is that every line g which intersects the convex hull of but avoids any of its points determines two non-empty subsets of <m\ S, say, and 5C,

SKJSC

= *n.

The two sets are separated from each other by the line g. Therefore for every B e r(m) we can find sets Sz c m such that B can be expressed as a disjoint union

B = U ««,

(5-2.1)

I

where at = [g e G : g separates St from Sf}. We can now apply Crofton's theorem and find /i(a^) by taking Dx and D2 to be the (polygonal) convex hulls of St and S,c. Using the notation of fig. 5.2.1, Mai) = l<*il + I < * 2 | - £ K I .

(5.2.2)

Clearly the existence of linear representations of the type (5.1.3) for general B e r(m) follows by simple addition of this expression written for all atoms participating in (5.2.1). But what about the algorithm for the coefficients?

5.3 Model approach and the Gauss-Bonnet theorem

103

Figure 5.2.1

The simplest solution is the following: we check that (5.2.2) coincides with the expression for ju(ak) given by (5.1.2)—(5.1.3). Then we use the additive property:

h i t 7') = l 4 1 ( t 1) whenever B is given by (5.2.1). The additivity follows from the definition of IB(( i,i j). j) This completes the first proof of (5.1.3). We mentioned in §5.1 that (5.1.3) can be derived from (5.1.1) by means of a direction-erasing map. To prove (5.1.1) we can now use the fact that it is possible to proceed in the converse direction. In fact, (5.1.1) follows from (5.1.3) by invariance of the measure / i o n G with respect to inversion of the line directions.

5.3 Model approach and the Gauss-Bonnet theorem The purpose of this section is to show that the decomposition (5.1.3) is based upon combinatorial relationships between certain subsets of G. It turns out that sets of the type [v] = {g e G : g hits a linear segment v c M2} are of particular importance (for the reasons explained in §5.12). (In [3] the sets [v] were called 'Buffon sets'.) We will use a model for G (i.e. a representation of G by a surface) in which each bundle = {geG:g

hits a point 9 e U2}

is a geodesic line. This will enable us to establish, as a by-product, connections between (5.1.3) and some well known facts of differential geometry. First consider the cylinder C = § ! x R representing the space G of oriented lines (see §2.1). On C each bundle [^] is represented by an ellipse, namely the trace on C of a plane through the point 0 which lies on the axis of C at the

104

5 Combinatorial integral geometry

Figure 5.3.2

p = 0 level. We map C into S 2 , the unit sphere with center O in !R3, by means of projection through 0, see fig. 5.3.1. This map is one-to-one except for the south and the north poles Q 1,Q2 e S2 which we remove. It is clear that each [^] is the trace on § 2 of a plane through 0, i.e. [^] is a great circle. Now the map which reverses the direction of the lines sends each point (cp, p) ofCto(

= E2\e,

where E 2 is the elliptical (or projective) plane, and Q corresponds to the pair 61,62. Next we ask what is the form of sets from r(m) in this model? We have seen that each [^] is a geodesic line. If ^ / 0>2 then [ ^ ] and [^ 2 ] are two geodesies intersecting at a single 'point', namely the line through ^ and ^2 (fig. 5.3.3). These divide E 2 into two connected components (called lunes): one such component is the needle set [v] where v = ^>1^>2. Since [v] is bounded, i.e. [v]

5.3 Model approach and the Gauss-Bonnet theorem

105

Q

Figure 5.3.3

Figure 5.3.4

should have a compact closure within E 2 \ Q , we identify [v] as the lune which does not cover the removed point Q. Given a set m = {^}", we have n bundles [ ^ ] in G, and the corresponding geodesic lines split E 2 into non-overlapping geodesically convex polygons (elementary polygons). They actually represent the atoms of r(m) except for the elementary polygon which covers the point Q on E 2 : it corresponds to the unbounded atom. Consider the simplest case, involving three points ^ , ^ 2 and @>. Consider the atom [v x ] n [v 2 ], where vx = &>
(5.3.1)

can be visualized directly from fig. 5.3.4 as a property of the demarcation of E 2 by three geodesies. Of course in (5.3.1) we ignore points which belong to the boundaries (i.e. lines from the bundles [ ^ ] , [ ^ ] [^ 2 ]). Let v be a vertex of the geodesic convex polygon D c E 2 . Two lunes are determined by the (two) continuations of geodesic sides meeting at v. That which contains int. D is termed the covering lune, while the other is the outer lune of D at v. Again, each of these lunes can be either unbounded or bounded, depending on whether it covers the deleted point Q or not.

106

5 Combinatorial integral geometry

If T c E 2 is a geodesic triangle and ll912 and /3 are the covering lunes of T, then the symmetric equation 2/T = Ilx + Ih + /,3 - 1

(5.3.2) c

an

holds. In particular if /x = [v x ], /2 = [v 2 ] and /3 = [ ^ i ^ 2 ] ( unbounded lune) we recover the decomposition (5.3.1). Integrating (5.3.2) with respect to the area measure a on E 2 yields the celebrated Euler formula for the area of a spherical triangle: fl(T)=|/1| + |/2l + | / 3 | - ^ where ^a(lt) = \lt\ is the interior angle of the lune. At the same time, integration of (5.3.1) with respect to the fVO2-invariant measure / i o n G yields or in a weaker form M + |v2|-|^2l^o, where | • | now denotes length in U2. We see that the usual triangle inequality has combinatorial roots in common with the Euler formula. We now return to the general case of a bounded geodesic convex polygon D on E 2 (an atom). By the triangulation {TJ of D shown in fig. 5.3.5 we have j

Substituting (5.3.2) for each triangle yields

2iD{g) = YJiii{g)-N, i

where {/J is the set of covering lunes of D and N is the number of triangles in the triangulation. A covering lune /,- can be unbounded. In case it is we substitute

Q

Figure 5.3.5

5.3 Model approach and the Gauss-Bonnet theorem

107

where c denotes the complement and obtain

2ID(g)=

I

lu{g)-

li are bounded coveringlunes

I

Iiltr(g) + s-N.

lt are unbounded coveringlunes

where s is the number of unbounded covering lunes. Two remarks are important. (a) If l( is unbounded and, in our triangulation, lt = \Jbk, where bk are covering lunes of the triangles x-p then exactly one of these lunes bk is unbounded (i.e. covers Q). (b) A bounded triangle always has exactly one unbounded covering lune. It follows from (a) and (b) that s = N, and we obtain an expression for JD, the indicator function of an atom D in terms of bounded lunes. Recall that only bounded lunes have the interpretation of sets [v]. The result is valid up to lines g which belong to a finite number of bundles:

2 / D = Z Kbounded covering lunes

Z k-

bounded outer lunes

(5-3-3)

Equation (5.2.2) can be obtained from (5.3.3) by integration with respect to the measure /i, recalling that M[v]) = 2|v|. The bounded covering lunes correspond to the segments dx and d2 (and it follows that D always has exactly two bounded covering lunes). Thus we have given a new proof of (5.2.2) and with it of (5.1.3) and (5.1.1) (see the end of §5.2). Yet (5.3.3) is of much greater value as explained by the following two remarks. First, (5.3.3) can be integrated with respect to any locally-finite measure m (assuming m([^]) = 0 for every bundle of lines [^]). After summation over atoms composing B we obtain m(B) = \ X c o .(B)m([^.]),

(5.3.4)

where B is the general set from r(m), [^.^.] = lines which separate &{ from ^.. Thus (5.1.3) has a very simple generalization to non-invariant measures, and, what is important, the coefficients ctj(B) do not depend on the choice of measure m (in fact they are topological invariants, see [3]). Secondly, if we rewrite (5.3.3) completely in terms of outer lunes (which can be unbounded) and integrate the result with respect to a (finite) area measure on E 2 , we obtain the classical Gauss-Bonnet formula for convex polygons on the elliptical plane. Thus (5.3.3) is a bridge which connects (5.1.3) and (5.1.1) with well-known facts of differential geometry.

5 Combinatorial integral geometry

108

5.4 Two examples Given a finite set m = {^} the collection of all segments ^ ^ is called a companion set of needles (needle = line segment). For brevity we denote the needles from the companion set as vs. In (5.1.3) the range of summation is usually less than the complete companion system. This range actually defines the skeleton of a set B e r(m). The skeleton consists of all needles vs = &{&* for ±± which IB( U j) depends on both of its binary arguments, + and - . Note that if vs belongs to the skeleton of B, then the continuation of vs (which is a line in U2) necessarily belongs to dB, the topological boundary of B. I As an important application of (5.1.3) and the four indicator formula, let us give the answer to J. J. Sylvester's question [23]: given a set {<5J of needles, what are the integer coefficients (5.1.2) when

B= H [«

or

B = (J [SiJf!

Let m, = ffi} be the set of endpoints of the needles St with which we start. We then construct the companion set of needles {vs}. Since we assume that no three points from m are collinear, the needles of the companion set {vs} may be divided into three categories: (1) the needles 5t of the original set; (2) needles dk joining two different needles 8{ and Sj9 such that 8t and Sj lie in different halfplanes with respect to dk; (3) needles sk joining two different needles 3t and Sj which lie in the same halfplane with respect to sk. We introduce a function /fc(v) defined on needles as follows: /k(v) = 1 if the continuation of 0 hits the interior of exactly k needles from the set {<5J, = 0 otherwise. We assume here that, for the case in which v = Ss, the needle Ss is not counted. The coefficients cs(B) for the two cases considered by Sylvester are summarized in table 5.4.1.

Table 5.4.1 B

\

nra

dk

*k

K-2

-h

-In-2

h

5.4 Two examples

109

Thus the representations (5.1.3) take the form =

2

I In-l{St)\Si\ + Z W ^ M I " I W ^ W i=l

k

=2 E EW W

(5.4.1)

k

- Z I0(dk)\dt\ + £ /o(s*)|s*l-

(5A2)

i=l

II The second example gives a solution to another problem by J. J. Sylvester, also posed (but not solved) in his paper [23]. Let a system Dl9 ..., Dn of bounded convex domains be given in U2. We consider the sets and }Dh Z)j< = {g e G : g separates Dt from Dj}. Define the Sylvester ring Sr{Dt} to be the minimal ring of subsets of G which contains all the sets [Df] and }Dh Dj{. The problem is to find the invariant measure of the elements of Sr{Dj. In order to established connections with §5.1 we first assume that the domains Dt are polygonal. Let m = {^} be the collection of vertices of polygons Dt. We have the basic relation

Sr{Dt} c r(m). Therefore, if the set {^} is non-degenerate, we may use (5.1.3) to find decompositions for every BeSr{Dt}. The characteristic feature of these decompositions will be the restriction of skeletons to a certain subclass of 0*^ needles. In particular, the diagonals of the polygons never occur in a skeleton of a B e Sr{D(} and so do the pairs 0*^ with ^ e dDh 0) e dDj if the continuation of 9^ hits the interior of Dt or Dj. The needles that remain are (1) 0>t and 0>j are endpoints of the same polygon side; (2) &{ and ^ are endpoints of a double support line (see fig. 5.4.1). For brevity we call type (2) segments 'strings'. In the case where at least some of the D{ cease to be polygons, a natural limiting procedure may lead to a decomposition of n(B) for B e Sr{Dt}.

d type

* ^ ^ / '

s type

Figure 5.4.1 The two types of strings

5 Combinatorial integral geometry

110

In this limiting process each (non-polygonal) Dt should be replaced by an approximating polygon D\n) (say D\n) a /).). The set Be Sr{Dt} should be replaced by the corresponding B(n) e Sr{D\n)}. The decomposition for jx(B{n)) should be constructed and its limit (as n -• oo) found. In this way the following proposition can be established. Assume that for the system of convex domains {Dt} no lines support more than two domains Dt simultaneously. Then for every B e Sr{Di} the invariant measure [i(B) can be written in the form

c,(B)dl+ dD{

T cv(B)\v\,

(5.4.3)

strings

where / denotes a point on 3D, d/ is a boundary length element, v is a generic notation for a string. Both functions ct(B) and cv(B) can take only values 0, ± 1 or ±2. Since we now do not exclude the fact that the boundaries of the D,'s can have curved parts, the strings in (5.4.3) can be as shown in fig. 5.4.2. For concrete sets B the functions ct(B) and cv(B) can be written in explicit form. Let us consider the case B = {g G G : g hits at least one of the IVs}In this case Cl(B)

= Jo(0,),

cy{B) = I0(gv) = -/ o (0v)

if v is of s type if vis of rf type,

where I0(g) = 1 if the number of the domains Dt whose interiors are hit by g is zero, = 0 otherwise; gl is the tangent line at the point / e dDt; gv is the continuation of the string v;

5 type

d type Figure 5.4.2

5.5 Rings in E

111

and the s and d types of a string are shown in fig. 5.4.2. The proof is by polygonal approximation.

5.5 Rings in E Let m = {^i} be a finite set of points in U3 with no three points on a line. Each plane e eE which does not pass through a point from m produces a separation of *n in two subsets. Two planes el9 e2 e E are called equivalent if they produce the same separation of m. Sets of equivalent planes are called atoms. All atoms are bounded, except for the one which corresponds to the separation 0 , m. We denote by r(m) the minimal ring of subsets of E which contains all bounded atoms (the Radon ring in the terminology of [3]). In this section we formulate the main combinatorial result for general bundleless measures m on E, i.e. for measures satisfying the condition ML^l) = 0 for every & e (R3, where [^] is the bundle of planes through &. The result is formulated in terms of 'wedges'. In the present context the term 'wedge' was first introduced in [3] although the corresponding notion has existed anonymously in integral geometry for a long time. A wedge is a pair W = (v, V), where v is a finite line segment in IR3 while V is an open domain in U3 bounded by two planes through v, called the faces of W. V consists of two disjoint parts forming a flat vertical angle (fig. 5.5.1). A wedge W = (v, V) is said to belong to the companion system of the set if (a) vis of the form (b) the interior of V does not contain any point of (c) on each face of W there lies, besides the endpoints of v, at least one other point from Thus the companion system always consists of only finite numbers of wedges. Given a measure m on E, we define the following 'wedge function' | W\: QL(e)m(de).

(5.5.1)

Here [v] denotes the set of planes which hit v, and oc(e) is the angle of the planar domain e n V (the angular trace of V on the plane e).

Figure 5.5.1

112

5 Combinatorial integral geometry

We have the following proposition. Let m be a bundleless measure on E. For every B e r(m) the value m(B) can be calculated as a sum

YJ

(5.5.2)

where cs(B) are integers which do not depend on the choice of the measure m, and the sum is extended over the companion set of wedges. The algorithm for the calculation of the integers cs(B) resembles that given by (5.1.2). Let the wedge in question be Ws = (vs, J^). Any plane e0 containing vs and passing within the domain Vs belongs to the boundary of four atoms, which we denote as + +

(u ;),

- -

(U j),

+ -

(U j \ ,

- +

(U ;)•

These refer to the four possible positions of the endpoints ^ and ^ or vs with respect to small displacements of e0. That is, + + and correspond to two cases in which both ^ and ^ remain in one halfplane; H— and —(correspond to the two cases in which ^ and ^ fall in different halfspaces (compare with fig. 5.1.2). We have

cs(B) = IB(l ]) + IB(l )) - IB(l }) - IB(l ] \

(5.5.3)

± ± where IB( i, j) are the values of the indicator IB(e) on the atoms. A complete proof of (5.5.1)—(5.5.3) can be found in [3], and we now outline its main features. The main idea is to reduce the problem to the planar result (5.1.3). This is done in the following way. Fix a line co through the origin O GU3 and choose a plane e(a>, 0) containing co. For each O e (0, n) let e(co, G o , where G o is the family of lines lying in the plane e(co, Q>). This furnishes a parametrization of E\II: to every pair (
5.6 Planes cutting a convex polyhedron

113

m(de) = /(
f jBn JB

The basic observation is that, given a collection m of points in [R3, B e r{m)

implies

i ( 5 n E o ) e r(^°),

where the set m* is the perpendicular projection of m onto the plane e(co, O), and r(^°) is the ring in G^ which corresponds to W*\ So for each set B e r(m) we can apply (5.3.4) to the inner integral. Another crucial step is to observe that the integer coefficients which solve the planar problem on each e(co, 3>) coincide with cs(B) as given by (5.5.3) (because the dependence of a planar coefficient on O reduces to dependence on a wedge). The final result (5.5.2)(5.5.3) follows by eliminating the dependence on a> by means of averaging with respect to dco/2n (the uniform distribution on S 2 /2). In the following we use several times a rather special case of (5.5.2)—(5.5.3) which we consider in the next section (this case admits an elementary complete proof). A general approach to the calculation of the function | W\ is considered in §5.8.

5.6 Planes cutting a convex polyhedron Let D be a bounded convex polyhedron in (R3, m be some bundleless measure on the space E, and let B = {eeE:e

hits D}.

Clearly this B belongs to r(m), where wi is defined to be the set of vertices of D. Therefore the value of m(B) can be calculated by means of (5.5.1)—(5.5.3). For this special B, however, there exists a very simple independent derivation of the same expression. We start from the observation that, for almost every plane which hits D, the intersection D n e is a bounded convex polygon whose vertices correspond (in a one-to-one manner) to the edges of D actually hit by e. We write the elementary fact that the sum of outer angles of D n e equals 2TC in the form of an identity between indicator functions: £ /[fli](e)a;(e) = 2nIB(e).

(5.6.1)

Here [ a j is the set of planes hitting an edge at of D. Hence at(e) is the angle of the trace left on e by the outer wedge constructed on a(; summation is by all edges of D. By definition, for an outer wedge the domain V (see fig. 5.5.1) does not possess points in common with the interior of D.

114

5 Combinatorial integral geometry

It remains for us to integrate (5.6.1) with respect to m:

~ 2n Jw "'^ m =

E

\W.\-

(5.6.2)

outerwedges onedges

We show in §5.8 that when m equals \i, the M 3 -invariant measure on E, we have the following wedge function: \W\=\\v\'\V\,

(5.6.3)

where |v| denotes the length of v and | V\ denotes the opening (flat angle) of the domain V. The results (5.6.2) and (5.6.3) together yield a classical formula: £ edgesofD

It can be instructive to obtain (5.6.2) by a direct application of the algorithm (5.5.2)-(5.5.3). We mention in this connection that the general algorithm is capable of solving a much more difficult problem concerning polyhedrons, namely that of finding the distribution of the number of vertices in a polygon which arises when D is sectioned by a random plane, see [3].

5.7 Reconstruction of the measure from a wedge function The wedge functions \W\ as introduced by (5.5.1) actually provide an alternative way of describing measures on the space E. Indeed, (5.5.2) shows that with the knowledge of | W\ the problem of calculation of m(B) is reduced to the linear combination of certain values of | W\. Of course there is a restriction that B should belong to r{m) for some m. However, the class of such sets B is a measure-determining class. Below we show how to reconstruct the density of a measure (assuming that the density exists) in terms of its function | W\. Let 8l9 S2 and <53 be three parallel non-coplanar segments in R3. We put B = {eeE:e

hits each 8h i = 1, 2, 3}.

To write (5.5.2) for this B we introduce the following notation. Let ei be the plane containing 8j and 8h i ^y, /. In the plane ei we consider the segments S (D

5(2)

J(l)

J(2)

shown in fig. 5.7.1. For each segment v = s|k) or df\ i = 1, 2, 3, k = 1, 2, we define the domain Vv by the following requirements: (a) Vv is bounded by two planes through v; (b) these planes pass through the endpoints of 8t; (c) Vv contains the segment 8t.

5.8 The wedge function in the shift-invariant case

115

Figure 5 J.I We assume that cos v£ = 1, i = 1, 2, 3 We consider the wedges from the system accompanying the set m = {endpoints of 8h i = 1, 2, 3}. The values of cs(B) on all wedges which are not of (v, Vv) type are zero. On the other hand, if Ws = (v, Vv) we have cs(B)= 1 = —1

if vis of df] type, if v is of s|k) type.

Thus the expression for m(B\ (5.5.2), has the form (5.7.1) where I is the set of six wedges of the type (df\ J^o); II is the set of six wedges of the type (s\k\ V^o). Let us now assume that the measure m possesses a density u with respect to the M 3-invariant measure de, i.e. m(de) = u(e) de. Our purpose is to find u(e) for some fixed e e L W e now choose the segments (5X, S2 and S3 to have a common infinitesimal length / and we let each St hit the plane e. According to (3.12.2) as / -• 0 we have Thus This, together with (5.7.1), solves our problem. We stress that there are many degrees of freedom with which the segments 5U S2 and (53 satisfying the abovementioned conditions can be constructed.

5.8 The wedge function in the shift-invariant case How can the function | W\ given by (5.5.1) actually be calculated? We give the answer for the T3-invariant measures on E, a case that is especially important

116

5 Combinatorial integral geometry

because of the connections with zonoids. Note that, because of the factorization of §2.9, all T3-invariant measures are bundleless and for them (5.5.2) applies without exception. Now let ju be such a measure. It is convenient to start with the case in which the rose of directions m of ja is concentrated at one point, i.e. is of d type. In symbols m = dw for some spatial direction co. Let W = (v, V) be a wedge (see fig. 5.5.1). For a <5-type m, (5.5.1) yields = \v\'F(V), where cov* is the angle between co and the direction v* of the segment v, and em is the plane orthogonal to co. The formulae of spherical trigonometry enable us to write an explicit expression for the angle a(ew). But now we will be interested in the following observation. For a fixed direction v* of v, each V defines a pair of arcs on the circle S t : they are the trace of V on S x placed in a plane perpendicular to v and centered at a point of v. Therefore, for fixed v*, F(V) is a function defined on the pairs of centrally-symmetrical arcs on S l 5 or simply on arcs of E x (see §1.5). Actually F(V) is a measure on E x (equivalently on (0, n)) which has a density 1 sin2 c. In Here c = c(co) is the angle between v* and the trace of ew on the plane e o (seefig.5.8.1): eo belongs to the bundle of planes through v, and 0 e E x denotes the angle of rotation around v*. Proof Applying standard formulae of spherical trigonometry we find that, if the opening of V is dv (infinitesimal), then

Figure 5.8.1

5.8 The wedge function in the shift-invariant case

117

where fi is the flat angle between e^ and the plane shown in fig. 5.8.1 The relation cos(co, v*) = sin c sin /? completes the proof. It is now straightforward to show that, for a general rose of direction m, the density of the corresponding F is expressed by the integral sin 2 c m(dco).

p(v*, 3>) = —

(5.8.1)

2;r J Finally, the wedge function is written in terms of p as follows: \W\ = |v|

p(v*,O)d<&.

(5.8.2)

The space where the function p(v*, <X>) is defined is the space F of flags. By definition, aflagf is a figure consisting of a line through 0 and a plane through this line. Therefore / = (n,O),

(5.8.3)

where Q e E 2 , O e El9 (see §1.5). On F there is a topology whose description resembles that of W3 (see §3.2). Note that topologically F is different from the product E 2 x E x ; however, this product can be used as a parameter space to represent F (in fact E 2 x Ex is a measure-representing model of F). We can represent flags in a way dual to (5.8.3) / = (co, q>\

(5.8.4)

where co e E 2 corresponds to the plane of the flag, cp e Ex represents the direction of the line of the flag as a direction on the plane co. We will see below that the possibility of dual representation of flags plays an important role in questions related to zonoids. Example When \i is M 3 invariant then m(dco) = dco and p reduces to a constant C. Then, according to (5.8.2), \W\ =

C-\v\'\V\.

To calculate C we take \V\ = n. By (5.5.1) | W\ will equal one-half the value of fi on the set of planes hitting v, i.e. (TT/2)-|V|. Hence, C = 1 / 2 and (5.6.3) follows.

118

5 Combinatorial integral geometry

5.9 Flag representations of convex bodies Of special importance is the wedge density Sin2 C

Ps = 27i

'

where we now consider the angle c (see fig. 5.8.1) as a function of flags: c = c(£,/) = c(£,Q,O) = c(£,a;,
(5.9.1)

3

Let D be a convex domain in U and let [D] be the set of planes hitting D. Then a direct integration of (5.9.1) yields ^([D]) = b(a

(5.9.2)

the value of the breadth function of D in the direction £. If D is a polyhedron, (5.6.4), in conjunction with the previous results, yields

I

outer wedges ;rofDwedges

W[ J K

= lsin 2 c(^/)m D (d/).

(5.9.3)

Here raD is a measure in the space F concentrated on the union of the sets {(a?,®):®6^}. is uniform in O on each of this sets, and Therefore the total measure is given by 1

w

~ ^.^I-IK-I

(5.9.4)

(compare with (5.6.4.)). The equality (5.9.3) is our flag representation for polyhedrons. The existence of similar representations for every bounded convex body D a [R3 follows now by weak convergence arguments. Namely, for any bounded convex D <= [R3, we can (and do) construct a sequence of uniformly bounded convex polyhedrons Dn such that lim bn(^) = b(£) (uniform convergence) n-*co

where bn and b are the breadth functions of Dn and D, respectively. In the flag representations (5.9.3) for Dn's the total measures of mD\ remain bounded (follows from (5.9.4)) and therefore a weakly convergent subsequence mn

—> m

5.10 Flag representations and zonoids

119

can be chosen. There is a complication in that the function sin2c is not continuous at £ = co. Therefore at first we can state only that the equation n2 c(ij)m(df)

(5.9.5)

holds for all values of £ except for those for which m({f = (co, (p):w = £}) > 0, i.e. except for countably many values of £. However, a further analysis shows that this restriction can be lifted, and in fact (5.9.5) holds identically for every

I We call (5.9.5) a flag representation of the breadth function of a bounded convex D a U3. If D is centrally-symmetrical, then we say that (5.9.5) is a flag representation of the body D itself Again we stress that in contrast with (2.10.2), which holds only for zonoids, flag representations are valid for all bounded convex centrally-symmetrical bodies in M3. Given D, the measure m in (5.9.5) is not determined in a unique way. However, flag representations may shed new light on some facts of the theory of convex bodies. For instance, Blaschke's basic principle of compactness of sets of uniformly bounded convex bodies now reduces to the similar property of measures with respect to weak convergence. In the next two sections we show that flag representations can be applied in order to obtain previously unknown facts.

5.10 Flag representations and zonoids Zonotopes are zonoids generated (via (2.10.2)) by purely atomic measures. This means that the breadth function of a zonotope admits representation of the type

b(S) = £\cos£ii\'qi9

(5.10.1)

i

where Qt e E 2 are some directions in U3, and qt are some positive weights. A zonotope is always a bounded centrally-symmetrical convex polyhedron. A bounded centrally-symmetrical convex polyhedron D is a zonotope if and only if (a) the collection of edges of D consists of classes such that in each class the edges are parallel and of equal length; (b) within each class the openings of outer flat angles by the edges sum up to27L The proof that (a) and (b) and (5.10.1) are equivalent is done by applying the notion of Minkowski's addition. In fact, (5.10.1) is the breadth function of

120

5 Combinatorial integral geometry

Minkowski's sum of n segments whose directions are Qt and whose lengths are qt. Hence (a) and (b). Conversely, suppose the conditions (a) and (b) are satisfied for a given polyhedron D. We write the flag representation (5.9.3) in the form

1

X classes

= £

edges from a class

qj\cos£Qjl

classes

where qj is the common length of the edges in a class, and Qj is their common direction. From the point of view of flag representations the above has the following interpretations. We have a set relation (J

{(Q, 0>): Q = af, O e V{} « a* x E x .

edges from a class

It follows from the remarks in §5.9 that on each of the above sets the measure mD is q} times the uniform measure on E x . We come to the conclusion that any zonotope D admits flag representation by means of a measure on F whose image under the map (5.8.3) is a factorized measure on E 2 x Et with a uniform factor on E x . What about flag representations for general zonoids? It is well known that a breadth function of any zonoid admits pointwise approximation by breadth functions of zonotopes. This means that the measure m in a flag representation (5.9.5) of a zonoid can be obtained as a weak limit of measures satisfying the factorization property. The limit will necessarily be of the same factorized type. Conversely, if the measure m in (5.9.5) is of this special type, then a direct integration reduces (5.9.5) to (2.10.2). Hence we have the following result A bounded convex centrally-symmetrical body D cz R3 is a zonoid if and only if its breadth function b(£) admits a flag representation by means of a measure on F whose image under the map (5.8.3) is a factorized measure on E 2 x Et with a uniform factor on E x ; the factor on E 2 coincides with the measure m in the representation (2.10.2) of a zonoid.

5.11 Planes hitting a smooth convex body in IR3 In this section we derive an explicit flag representation for a general smooth (non-polyhedral) bounded convex body K a R3. The 'density' of the representing measure is expressed in terms of curvatures.

5.11 Planes hitting a smooth convex body in R3

121

We start with the identity •2nI[K](e)

(5.11.1)

which expresses the familar fact of planar geometry that as we move along the boundary of a bounded convex domain the tangent line rotates by an angle 2%. Here [X] cz E is the set of planes e which hit K, &(e, I) is the curvature at the point ofdKne determined by a longal coordinate /, I[K](e) is the indicator function of the set [X]. We integrate (5.11.1) over the space E with respect to the measure fi{de) = <^(dco) dp (equivalently, over the bundle of parallel planes orthogonal to the spatial direction £ with respect to linear Lebesgue measure dp). We get the following expression for the value of the breadth function at £:

For a fixed £ each pair (p, I) determines a point 0* on dK. By d& we denote an elementary rectangle at 9 which lies on t(0>) - the tangent plane at 9 (d& also denotes the area of the rectangle). Let the sides of the rectangle be d/ and dh so that

d» = df dh. Let y be the angle between dh and dp (or £). We have dp = cos 7 dh and therefore d/ dp = cos y d0>. The next step is to express the curvature &{&> p, /) in terms of the main curvatures a>1 and az2 at ^ \ Recall that ccx and a?2 are two extremal values of the normal curvature &n () and leaves a linear trace in the direction cp on We will make use of a Euler formula [6] ccn((p) = azx COS2 Pxq> + where j5x and /52 are (mutually perpendicular) 'main' directions in We will also use a formula which gives the value of the curvature at 0> of the traces of dK on the planes which are no longer perpendicular

to t{&\ Denote by #(^, cp) a line on t(0>) which contains &> and has (planar) direction cp. Let e(£P, q>9 y) be the plane through g(0>, cp) rotated around g{^ cp) by an angle y. We assume that

122

5 Combinatorial integral geometry

For the curvature ^ ( ^ , cp, y) of e(^, cp, y) n dK at 0> we have (Meusnier's theorem) a>(&, cp, y) = (cos

yy^^cp).

The two classical results in our situation yield a>(t P, I) = &(&, <(>& I) = (cos y ) " 1 ^ ! cos 2fi^cp^+

<JC2

cos 2 /?2<^]

where ^ is the direction of the trace on t(&) of a plane orthogonal to <J. The expression for b({) now takes the form b(i) = — I cc(i, p, l)cos y d0> vx COS 2 Pi(P£ 4- &2 COS

To obtain the desired flag representation we convert the above to integration over a sphere by means of d ^ = rx r2 da> where r( = l/&i9 i = 1, 2 are the main curvature radii, da> is the area measure on S 2 . We get finally 2 Hi) = ^~ f [rxx sin si ^

+ r2 sin2 ^ ] dw

(5.11.2)

This is equivalent to a flag representation offr(£)since the integrand coincides with the result of integration of sin2c with respect to the measure riSfil(d(/>) + r25fi2(dq>)9 We note that (5.11.2) is a translational counterpart of the much wider known relation referring to the M 3 -invariant measure )U0

=f

\[

(5.11.3)

^ J JEE 2 The expression on the right-hand side is called the 'integral of mean curvature'. Clearly (5.11.3) follows from (5.11.2) by averaging over directions ^. We note also that, for a general T 3 -invariant measure /z, the expression of/z([X]) in terms of corresponding p can be obtained by integrating (5.11.2) with respect to the rose of directions of//:

where the flags /£ are defined in terms of (5.8.4) as ft = (orientation a> of the tangent plane, main direction

5.12 Other ramifications and historical remarks

123

5.12 Other ramifications and historical remarks The idea of introducing measures into the space of lines in the plane was already implicit in Buffon's classical needle problem [36]. Let us recall its formulation. The plane is ruled by a fixed lattice of parallel lines a unit distance apart. A needle v of length |v| < 1 is 'thrown' at random onto the plane. What is the probability p of the event that in its final position the needle will be intersected by a line of the lattice? In an equivalent formulation, the needle and the lattice exchange roles and one assumes that it is now the needle which is fixed in the plane with the lattice being thrown down at random. Without loss of generality one may assume that the needle lies within some fixed open disc D of unit diameter. Then in all possible outcomes of the lattice-throwing experiment, the disc is intersected by exactly one line of the lattice if we assume that the case of tangency is 'impossible' (i.e. of probability 0). Since other lines of the lattice now play no role, we may fix our attention on this single line, gD say, intersecting D, and Buffon's original problem is now replaced by the following one: what is the probability that a random line gD intersecting D should also intersect the needle? We may refer to this as the dual problem to the classical Buffon's needle problem. It is clear that in Buffon's original problem the solution depends on how the experiment is performed, i.e. on the distribution of the final position of the needle with respect to the lattice. Analogously in the reformulated version the result depends on the distribution P of the random line gD. In the classical solution to Buffon's problem it is assumed that the center of the needle and its orientation are independently and uniformly distributed. This means that the projection of the center onto a line perpendicular to the lines of the lattice is uniformly distributed on some segment of unit length, and the angle between the line containing the needle and the lines of the lattice is independent and uniformly distributed on (0, n). With these assumptions, 2|v| This example is the earliest instance of the calculation of a 'geometrical probability'. To this result corresponds the following solution of the dual problem: there is a unique distribution P of the random line gD such that P{gD intersects v} =

2 -1 v| n

for every needle v within D. This distribution P is proportional to the restriction to the set of lines intersecting D of the M2-invariant measure on the space of lines in the plane (see §3.6). This direct connection with the classical Buffon problem provides justification of terming the subsets

124

5 Combinatorial integral geometry

[v] = {lines that hit v} as 'Buffon sets' see [3]. The use of other distributions for gD was first made by Bertrand [12] for the purpose of showing that the notion of 'random secant' admitted several interpretations and therefore allowed paradoxes ('Bertrand's paradoxes'). These could be avoided by adopting some principle leading to a unique choice among possible distributions, and it was considered that a 'natural' choice was one respecting the relation of Euclidean congruence, in that geometric events congruent to each other under Euclidean motions should have equal probabilities. This point of view is clearly expounded by, for example, Deltheil [22]. However, from a more modern point of view, interest in a general P has many other justifications. We mention only the simple observation that if P{gD hits ^ } = 0 for every point ^ e H 3 then the probability P{gD intersects the needle v}, considered as a function of the needle v cz D, is in fact always a continuous linearly-additive pseudometric on D (see §1.7). Remarkably, combinatorial integral geometry has provided (see [3], chapter 6) tools for proving that the converse is also true. This amounts to a combinatorial solution of the fourth of the D. Hilbert's famous problems, see the appendix by Baddeley in [3]. The first person to realize the possibilities of integration in the space G (and other spaces of geometrical elements) as a useful tool for geometrical investigation was Blaschke, who proposed the title 'integral geometry' as an appropriate name for the whole topic. But it was J. J. Sylvester who first glimpsed in 1891 (see [23]) the existence of decompositions (5.1.1) and (5.1.3) in an attempt to solve a problem, which in accordance with our above remarks may be considered to be a direct generalization of the classical Buffon problem and later became known as the 'Buffon - Sylvester problem'. It is as follows (compare with §5.4). In the plane, n needles v x ,..., vn are fixed in a general position. What is the invariant measure of the sets

H [vj

and

0 [vj,

where [v] = {g e G : g separates the endpoints of v}? Sylvester's result was that the invariant measures in question 'become Diophantine linear functions of the sides of the complete 2n-gonal figure of which the n pairs of extremities of the needles are the angles'. Of course, this was a loose expression of (5.4.1) and (5.4.2).

5.12 Other ramifications and historical remarks

125

Neither in [23] nor later was any practical algorithm proposed for finding the corresponding integers, and perhaps this is why the whole problem has been somewhat neglected. Finally the problem received a very simple solution,firstgiven in [28]. Soon it became clear that Sylvester's Diophantine decomposition principle is at the source of a potentially vast and fruitful theory. The elements for the planar case have been expounded in the preceding sections. The theory in many dimensions has an interesting relation to a question posed by Radon. He considered n points &{ a Ud in a general position and tried to calculate the number of different partition of {^} induced by hyperplanes. The answer obtained by many authors (Harding [24], Schlafli [25], Watson [26]) was that this number equals

If d is even, the above obviously equals

that is the number of odd-dimensional simplices 0 S , which have points from {^} for their vertices. Let [ 0 ] be the set of hyperplanes hitting the simplex 0. Clearly for every s [® J = U «r, (5-12.1) where ar are sets of equivalent hyperplanes: we call two hyperplanes equivalent if they induce the same partition of {^}. The relation (5.12.1) ignores bundles of hyperplanes through the points ^ . For any measure m in the space of hyperplanes for which m(any bundle) = 0, we have There is the fundamental result [29]: the matrix \\Srs \\ is square ifd is even, and in this case it can be inverted. From the resulting expression m(ar) = £ c w m ( [ 0 J ) (5.12.2) follows the existence of similar decompositions for every element Berffi}, the 'Radon ring' generated by the bounded atoms ar. An algorithm for calculating crs for d = 4 is given in [3]. The decompositions (5.12.2) have elegant implications in the geometry of the many-dimensional non-Euclidean elliptic spaces. The approach to these questions is converse to the one in §5.3: we reinterpret the facts concerning hyperplanes as facts of geometry in ellipitcal spaces. This yields a 'combinatorial' version of the Gauss-Bonnet formula for convex polyhedrons in many dimensional elliptical spaces (see [27]). It gives

126

5 Combinatorial integral geometry

the volume in terms of the openings of certain angles associated with the polyhedron (only angles having even-dimensional edges participate). The decomposition (5.12.2) does not survive for odd-dimensional Euclidean spaces. The situation for E (planes in U3) has been considered in detail in §5.5. The basic facts of the resulting theory of flag representations of convex bodies in U3 were first published by the author in [58]. See also [45], [62]. We also mention that the decompositions (5.4.1) served in [3] as a starting point for the derivation of various geometrical identities and inequalities concerning both convex and non-convex domains in (R2 and U3. We give an example of such a derivation in §6.9. In the approach to the stochastic geometry problems used in chapters 9 and 10 of [3], these identities have been crucial. However, in chapter 10 of this book we treat similar problems by a direct new method of averaging the basic decompositions of (5.1.3) or (5.4.3).

Basic integrals

This chapter presents several integrations aimed on the one hand at complementing the preceding material and, on the other, at providing tools for the study of geometrical processes in the chapters to come. The integrals in the first few sections refer to what can be called 'translationinvariant integral geometry': they generalize the basic formulae of classical integral geometry (as presented by Blaschke in [15], say). The section on translational analysis of discrete point sets, as well as a detailed treatment of an extremal property of the uniform distribution in the space of directions in §6.7, also belongs here. We also include here some integrals which belong to Euclidean motioninvariant integral geometry. They are related to the analysis of sets of segments by means of moving test segments, Pleijel identities, and related isoperimetric inequalities. We complete the chapter with the calculation of some integrals in terms of elementary functions. They refer to random chord length distribution for convex polygons as well as to triangular shapes.

6.1 Integrating the number of intersections Let tip be a T2-invariant measure on G whose rose of directions is a ^-measure concentrated on a direction /?, i.e. fifi(dg) = dp Sp(d
Let {<5J be a collection of line segments in R2, and let |<5f| and af be the length and the direction, respectively, of <5£. It follows (see §2.10) that

Summation over i yields

128

6 Basic integrals

I

^)\Pi,

(6.1.1)

where L = YJ \<>i\ is the total length of the segments; Pi = \di\ -L" 1 is the probability that a random point dropped on (J <5f (with uniform distribution on this set) will lie on a segment with orientation af; n(g) = £/[<5.](0) is the number of intersections of the line g e G with the segments. We call the integral

(

f \ [

(6.1.2)

the multiprojection of the set {<5J on the line perpendicular to the direction /?: each point p on this line is counted with multiplicity equal to the number of points from (J 5t which are projected into p. By using (6.1.2) we extend the notion of multiprojection to that for rather general curves i f on IR2: it is enough to define n(g) to be the number of intersections of i f with the line g. By taking the appropriate limit, (6.1.1) can be generalized to

*\ = L f IsinQS - a)|p(da)

(6.1.3)

valid for rather general curves if having finite length L. Here p(da) is a probability distribution on E 1? namely the distribution of the curve element direction at a point dropped at random (with uniform distribution) on if. Similar expressions can be obtained for multiprojections of curves or surfaces in U3. Let us first consider the case of a curve ^£ in U3. We consider its perpendicular projection on a line in U3 which has a direction co e E 2 . Let n(e) = n(a>, p)

be the number of intersections of the curve if with the plane e EE having coordinates (co, p\ see §2.7. Otherwise stated, n(a>, p) is the multiplicity of the projection at p. By definition, the multiprojection of if c: [R3 on a line with direction co is the integral

=

o, p) dp.

Applying the remarks of §2.10 and acting in the same way as before we find zx(co) = L

|cos(co, £)|p(d£)

(6.1.4)

first for broken lines in U3 and then for general curves of finite length L. Clearly

6.1 Integrating the number of intersections

129

p(d£) may be interpreted as the probability distribution of the direction of the curve element at a point dropped at random (with uniform distribution) on 1£. Thus p is a probability distribution on E 2 . Let £f be a surface in IR3. We denote by n(y) = n(o), 9) the number of intersections of Sf with a line y eT having coordinates (co, 9\ see §2.6. Equivalently n(co, 9) can be defined as the multiplicity of the projection at 9 on a plane having fixed normal direction co. Now the multiprojection is given by the integral z2((o) =

n{co, 9) d0>.

Using the remarks of §2.10 we derive the equation

z2(co) = S f|cos(a>,£)|p(d£)

(6.1.5)

first for finite unions of flats in U3 and then, by a natural passage to the limit, for rather general surfaces of finite area S. Here p is the probability distribution of the direction normal to ^ at a point dropped at random (with uniform distribution) on Sf. Again p is a probability distribution on E 2 . The integrals in (6.1.3)—(6.1.5) are of the same general form as in (2.10.1) or (2.10.2) but now they are produced not by measures or convex domains but by rather arbitrary curves or surfaces. It is in the spirit of stereology to ask what can be inferred about such objects in terms of their multiprojections given as functions of directions. The first thing that we conclude is that we can at most hope for the reconstruction of the probabilities p together with quantities L or S. Another conclusion is that multiprojections coincide with breadth functions of certain convex domains (zonoids if we are in IR3). The solution for the planar case was described in §2.11; we discuss the spatial case in the next section. Using the formulae of §§3.6, 3.9 and 3.11 we find that

J

(z(P)dfi= (n(g)dg = 2L,

(6.1.6)

z^co) dco =

n(e) de = TLL,

(6.1.7)

z2(co) dco =

n(y) dy = nS.

(6.1.8)

J

On the right-hand sides are classical integrals of the numbers of intersections with respect to M -invariant measures and their well known values (see [16] and [2]). Usually the above values are obtained by applying (3.7.3), (3.12.1) and (3.10.1). We have for instance

130

6 Basic integrals

r

r\dt\ r*

y[Si](g)dg=\

I sin ^rdx (1^ = 21^1,

and from this (6.1.6) follows (first for broken lines by summation). There are important corollaries of (6.1.6) and (6.1.8) referring to the case in which the curve or the surface in question happens to be the boundary dD of a convex domain D in U2 or U3. In such a case we have in (6.1.6) n{g) = 2 if g hits D

= 0 otherwise, and the integral reduces to two times the measure of the set Hence for the M 2 -invariant measure fi0 on G we have J M M ) = \8D\ = the length of the perimeter of D.

(6.1.9)

Similarly for the IVD 3 -invariant measure /n0 in the space T we find fio(y eT :y hits a convex D) = — times the surface area of D. (6.1.10) In the case of a polyhedral convex D c H 3 the M 3 -invariant measure of the set {eeE:e

hits D}

has been found in §5.6, and for smooth D in §5.11.

6.2 The zonoid equation It follows from the remarks of the previous section that certain problems in the field of'reconstruction from projections' are mathematically equivalent to the problem of solving the equation b{£)=

|cos£Q|m(dQ).

(6.2.1)

This requires the recovery of the measure m defined on the space E 2 in terms of the function b(£) defined on the same space. The existing work on (6.2.1) and its generalization to higher dimensions is considerable and is reviewed in [7]. Here we consider some basic facts only. I It is natural to consider the solutions of (6.2.1) in the class of signed measures. Within this class we have the uniqueness result: if £>(£) = 0 then (6.2.1) has only one solution, namely m = 0. We have seen in §2.12 that an m which has a density assuming both positive and negative values can produce a positive fc(^). In such cases b(£) always happens to be a breadth function of some bounded centrally-symmetrical convex body in R3. Do breadth functions always admit the representation (6.2.1) by means of signed measures? This question has an affirmative answer for sufficiently

6.3 Integrating the Lebesgue measure of the intersection set

131

smooth bodies. If additionally we require that the solution be a measure (rather then a signed measure) then the body should be a zonoid, see §§1.7 and 5.10. Thus the problem of the description of zonoids in terms of their b(£) values is essentially the problem of the description of the solvability domain of (6.2.1) in the class of measures in our usual sense. Many questions are still without answers. II Under the assumptions of smoothness of b(^) and the existence of the density m(dQ) = u(Q) dQ a formal solution of (6.2.1) can be based on the remarks of §2.12. Equation (2.12.9) expresses in terms of b(£) the integrals of the density function u(Q) extended over geodesies in the space E 2 . We face the problem first considered by Funk [5]: we know the integrals of a symmetrical function defined on § 2 extended over great circles; the problem is to find the function. As shown by Funk, this problem is easily reduced to the classical Abel equation. Another approach can be based upon the decomposition of b(£) in a series using spherical functions. Here we have the advantage that spherical functions are eigenfunctions of both operators L1H = 2H + A2H and

-J.

H d®.

Thus they are eigenfunctions of the product LJ 1 !^, see [6]. Since the corresponding eigenvalues are known (see any handbook on spherical functions), we can formally transform the series for b(£) into a series for u. As a rule, the convergence rate of the series for u is worse than that for b(£). This is the cause of the substantial complications in numerical computation. Here also we have many unanswered questions.

6.3 Integrating the Lebesgue measure of the intersection set Taking the Lebesgue measure of the intersection set can be preferable to counting the number of points. For instance, let ^ be a set on the plane and let Lx(g n J*) be the linear Lebesgue measure (total length) of the set g n $F. For any T2-invariant measure \x on the space G we have (6.3.1) where L2(^)

is the area (two-dimensional Lebesgue measure) of J5".

132

6 Basic integrals

Figure 6.3.1 The shaded area is dS

To prove (6.3.1) we factorize /i as in §2.9 and use the fact that g and dp are perpendicular; therefore an area element, see fig. 6.3.1. For each direction cp we have the constant value of the integral

hence (6.3.1). The constant cx depends only on the measure factor m and equals the total measure of m: c1=m(E1). 3

For a set 2F c R we have two analogs of (6.3.1). We can integrate the linear Lebesgue measure L^y n $F) of the intersection set with respect to a T3-invariant measure \i in the space F. Using the factorization of §2.9 and the fact that Lx(y n^) d£P is a volume element, we find that (6.3.2) J Here c2 = m(E2) where m is the measure determined by the factorization. Let L2(e n
c2 = 2TC,

C3

=

2TL

6.4 Vertical windows and shift-in variance

133

In contrast to this, the following result referring to the M 3 -invariant measure fi0 on E cannot be easily generalized to T3-invariant cases. Let 9* be a surface in R3 with area S. Let l(e) be the length of the curve 9 n e, e e E. We have l(e) de = 7i2S.

(6.3.4)

To prove (6.3.4) we integrate (6.1.8) over the circle S x with uniform density dcp\ n(y) dy dcp = 2n2S.

From (3.13.6) and (6.1.6) we conclude that n(y) dy dcp = 2

JJ

lye) de,

J

hence (6.3.4).

6.4 Vertical windows and shift-invariance We will consider a pair of'vertical windows' vx and v2 as shown in fig. 6.4.1. Let a line g0 be fixed on the plane. We consider the Haar measure of the set B = {M e M2 : Mg0 e [ v j n [v 2 ] and MO efc(r,0)}, where MX is the image of the set X under the notion M and b(r, O) is the disc of radius r centered at O. It is natural to use the formula (3.13.3). For this purpose it is necessary to choose a directed unit segment 50 a U2 and identify M with Md0. Then g and t in (3.13.3) denote the line which carries MS0 and the shift of Md0 along g. We stress that by the properties of a Haar measure (3.13.3) remains valid for any choice of <50. We now choose 60 as shown in fig. 6.4.1 and then find

H(B) = f J[v1]n[v2]

dg I dt jBg

where H denotes the Haar measure and Bg = {M

EM2:

Mg0 = g and MO efe(r,O)}.

This expression makes clear the asymptotic behavior of H(B) when we let the

X

2

Figure 6.4.1 The segments vt and v2 are perpendicular to the x axis. The source of <50 is the foot of the perpendicular from O onto g0, d = |xj — x 2 |, / = K | = |v2|.

134

6 Basic integrals

length / of the windows tend to zero. Under this assumption the set [v x ] n [v 2 ] shrinks down to the x-axis for every value of x l 5 x 2 . By the integral mean theorem we get

H(B) = [

dt)'[

\JBg.

J

dg

\JlVilnIv2]

where g* is some line from [v x ] n [v 2 ]. From (3.7.4) we find that

and

L r

lim f

dg = - + o(l2)

r

dt= f

'-^0 JBg.

dt = 2\x(g0)

JBOX

where BOx corresponds to the x-axis, x(Go) = Go n Hr> O). The last equality follows from the possibility of identifying BOx with two chords of b(r, O) parallel to the x-axis whose lengths are equal to locus of MO when M e BOx). We gather from the above that

where d is the distance between the windows. The important feature of this result is that lim l~2H(B) i-o

is invariant with respect to horizontal rigid shifts of the windows, i.e. it depends on x x and x 2 only through d = |x x — x 2 |. We will encounter a version of this effect in (6.8.10) in the analysis of systems of lines.

6.5 Vertical windows and a pair of non-parallel lines Let #! and g2 be two lines which intersect at a point and form an angle of opening a 0 . We again consider the vertical windows vx and v2 of the previous section but now the problem is to find the Haar measure H of the set B* = {M eM2: Mgx hits vl9 Mg2 hits v 2 }. There is an elegant way of calculating H(B*) which is based upon the identities (3.7.3) and (3.15.1). We write dgx = sin \j/1 dy1 d^/x dg2 = sin \\t2 dy2 d\j/2,

where yt and ^ are the coordinates of a line as in (3.7.3), with vt taken to be the reference line. Let

135

6.5 Vertical windows and a pair of non-parallel lines

/

Mg7

Figure 6.5.1

be the angle between the lines yu^i we get

and y2, *A2 ( s e e fig- 6.5.1). By multiplication

/(a) dgx dg2 = /(a) sin ij/1 sin fa dyx dy2 di//l di//2 (where / is some function) or, using (3.15.1), /(a) dM sin a da = /(a) sin xj/^ sin i//2 dyl dy2 di/^ d\jj2.

(6.5.1)

We can formally put in this equation /(a) = <5aoM(sin a)" 1 ,

(6.5.2)

where 4 o (a) is Dirac's ^-function concentrated on a 0 . By the usual rules of operating <5-functions

u<

dM da = H(B*).

Therefore we can find H(B*) by integrating the left-hand side of (6.5.1) with / given by (6.5.2). Integration by dyx dy2 yields

H(B*) = 2l2

40(a)(sin a)"1 sin fa sin fa dfa A fa.

The coefficient 2 appeared because the domain of integration (0, 1) x (0, 1) x (0, ri) x (0, n) corresponds to a half of B*. This follows from the observation that after rotation by n of the lines Mgx and Mg2 on fig. 6.5.1 around the point 9 we get the same values of yl9 y2, \j/l9 \j/2. One more integration can be performed due to the presence of the (5-factor after we replace the variables ij/1 and \j/2 by xjj^ and a. Since always we get H(B*) = 2r(sin a 0 ) *i f* sin ^|sin(^ — a 0 )

(6.5.3)

Jo A straightforward integration then yields the final result (6.5.4)

136

6 Basic integrals

Remark The above calculations remain valid for arbitrary pairs of parallel windows (not necessarily sides of a rectangle as in fig. 10.1.1). In particular the windows can lie on a line and even coincide (since H(B*) does not depend on the distance between the windows). This provides a check for (6.5.4) based upon a result of Santalo quoted below in (6.8.4). Adding up Santalo's expressions for i// = a 0 and \jj = n — a0 yields (6.5.4). If the windows are not parallel, then (6.5.3) becomes H(B*) = 2/2(sin o^)"1

sin ^|sin(^ - a 0 - P)\ cty,

(6.5.5)

where /? is the angle between the windows. Expression of this integral in terms of elementary functions is an easy modification of (6.5.4). There is an asymptotic counterpart of (6.5.3) for pairs of non-parallel segments. Let s1 and s2 be two such segments fixed in 1R2. The problem is to find the Haar measure of the set B** = {M EM2: MS1 hits vl9 Ms2 hits v 2 }. We can think of sx and s2 as being situated on the lines gx and g2 in fig. 6.5.1. Then H(B**) can be found by integrating the right-hand side of (6.5.1) with /(a) as in (6.5.2): H(B**) = o)" 1

dy1 J0

dy2 J0

£

IB**(yu y2,i//J)

sin i//\sin(\p - ot0)\ d\//

Jo »=1,2

Here the variables yl9 y2, \j/9 i determine a motion M, in particular the two values of i correspond to two rotations around the point

= \ * XX V*(0, 0, ^

lim r 2 sin OL0H{B**) = \

i) sin ^|sin(^ - a o )| d^.

(6.5.6)

J

Jo » = 1,

This expression is a bounded function of a 0 and other fMl2-invariant parameters which determine the segments sx and s2. Similar results can be obtained for a pair of discs of unit radius. On the set of discs whose circumference hits a window vt at one point we introduce the coordinates yh the usual one-dimensional coordinate of the intersection point on vh and ijjh the angle at which the intersection at y{ occurs. Because each disc is determined by its center Q e U2, the planar Lebesgue measure dQ determines a measure in the space of discs. On these sets we have dQi = sin ^ d^. dyt. By using (3.14.1) and applying the above method of ^-functions we come to the following result.

6.6 Translational analysis of realizations

137

Given two discs b1 and b2 on (R2, the Haar measure of the set {M GM2: Mbx hits vu Mb2 hits v2} 2

2

equals cl + o(l ) as / -> 0. The constant c depends on the distance between the windows and the distance p between the disc centers; p/l2 times this measure is a bounded function. Such results can be used in the study of probabilities of hitting of pairs of vertical windows in geometrical processes. The role of these probabilities will become clear from the contents of chapter 10.

6.6 Translational analysis of realizations Integration with respect to Haar measures on groups plays a significant role in many problems of stochastic geometry. In this section we consider several integrals for the group Tn of translations of Un. We apply them in the theory of Tn-invariant point processes in Un (in chapter 8). We denote the Haar measure on Tn either by hn or, if under the sign of integral, by At. We begin with the following simplest result. Assume that a point g? eUn and a domain D a Un are fixed. Then hn({tETn:0>etD})

= Ln(D%

(6.6.1)

n

where Ln is the Lebesgue measure in U . Proof We can assume that 0> = O, the origin in Un. If we represent shifts by points in Un according to the map

t->tO then the image of the set {t e Jn: O e tD} will be a domain D' a Rn obtained from D by means of reflection through the point 0. Thus (6.6.1) follows from the fact that Ln{D>) =

Ln(D\

and because the image of hn under the above map is just Ln. If we take D = b(r, O\ the ball of radius r centered at O, then the image of the set {t eJn:^ will be b(r, &\ N o w let a finite collection of points {^} be fixed in W. We define

e tD}

N(t) = the number of points from {^} within tD.

We have (6.6.2) where = 0 otherwise.

138

6 Basic integrals

Since (6.6.1) can be written as ( integration of (6.6.2) yields

r -sLn{D\

(6.6.3)

where s is the total number of points in As the number of points in {^} tends to infinity, so both sides of (6.6.3) tend to infinity. However, a modification of (6.6.3) yields more information than this. We will use the following notation:

N(m, D) = the number of points in <m n D, fc(r, O) = {t e Jn: tO efe(r,0)}. We will assume that for every r > 0 N(m9 b(r9 0)) < oo; in other words that m is a 'realization' in the sense of §7.1. Let us consider the integral N(m9 tD) dt. JE(r,O)

An asymptotic expression for this integral can be found if we assume that the domain D is infinitesimal. For simplicity let us take D = b(e, 0)

(6.6.4)

and assume that e->0. If <m has no points on the circumference of b(r, O) then N(m9 tD) dt = Ln(D) - N(m9 b(r9 O))

(6.6.5)

b(r,O)

for all sufficiently small values of e. Proof Let ex > 0 be the minimal distance of the points from m to db(r, O) and e2 be the halved minimal distance between the pairs of points from m n b(r, 0). For values s < min el9 s2 the function N(m, tD) has the following simple structure on the set b(r, 0): N{*n9 tD) = 1 = o Hence (6.6.5).

within the small circles shown in fig. 6.6.1 elsewhere in b(r, O). (6 6 6 )

6.6 Translational analysis of realizations

139

b(r, O)

Figure 6.6.1 The radii of the small circles are e; their centers are from the set m n b(r, 0)

Clearly (6.6.6) implies that for D, as in (6.6.4), \im(Ln(D)yl I

I^m, tD) dt = N(m, b{r, O))

(6.6.7)

Jb(r,O)

and

rC

°o

Jb( b(r,O)fcY = 2 klk{*n, tD) dt = 0.

(6.6.8)

Here and in what follows Ik(<m, tD) = 1

if the number of points in <m n tD equals k

= 0 otherwise. If we delete the condition that the domain D be infinitesimal (i.e. that D can be covered by a ball of infinitesimal radius) and retain only the requirement that limLM(D) = 0 then the last two relations do not necessarily hold. The sequence of rectangles in the plane D = {x, y:0 < x < 1,

0 < y < e}9

e->0

can be an example: in this case (6.6.8) collapses if m possesses pairs of points parallel to the x-axis. Nevertheless both (6.6.7) and (6.6.8) remain valid for the sequence of annuli

D = b(r + e, O)\b(r, 0),

e^O.

This follows from the fact that in this case for every two points ^ , ^ 2 {teJn:

0>l9 0>2 e tD} = t,Dn

G

^n

t2D,

where tt = O> 15

t2 =

2,

since the Lebesgue measure of the above intersection set is o(e), seefig.6.6.2.

6 Basic integrals

140

Figure 6.6.2 The annuli are ^ D and t2D\ their intersections are shaded

D

Figure 6.6.3

In the theory of point processes, yet another modification of (6.6.7) is of importance. Let us fix a collection D1,..., Ds of domains in Un (seefig.6.6.3), non-negative integers kl9..., ks and a 'realization' m. We consider the set

and as usual IA(t) = 1

= 0

iite

A

otherwise.

Let us denote by t( the shift which sends a point 0 the following two conditions are satisfied: (a) m n db(r, O) is an empty set; (b) for every ^ e ^ n ft(r, O) the set ^ ^ does not possess points on the boundaries of the domains Dx,..., Ds; then

f Jb(r,O)

I1(™,tb(e,O))IA(t)dt=

W- (6-6-9)

6.7 Integrals over product spaces

141

Proof By the remarks which led to (6.6.6), condition (a) implies

f

h (m, tb(e, O))IA(t) dr = X f

Jb(r,O)

IA(t) dt,

i Jb(e,-^)

where the point — ^ corresponds to ^ under reflection through O. Condition (b) implies that the function IA(t) has constant values within b(e, —0*i) for sufficiently small values of s, i.e.

IA(t) = lA(ti)

within

He,-^).

Hence,

L

and (6.6.9) follows. If condition (b) is violated then, in the limit of (6.6.9), terms depending on the properties of the boundaries of the domains Dt will also have to be included. We leave the details to the reader.

6.7 Integrals over product spaces We start with several integrals from classical integral geometry, i.e. we refer to M -invariant measures and their products. (1) The result of integration of (3.14.1) over the set {Ql9 Q2eD} =D x D, where D is a convex domain in R2, is as follows:

.I-

X3 Ag = 3(L2(D))2.

(6.7.1)

Here L2(D) is the area of D, x stands for the length of the chord g nD9 and integration is over G. We obtain a similar result by integrating (3.14.3) and using (3.13.2), namely

[•

X* dy = 6(L3(D))2,

(6.7.2)

where now D is a convex domain in R3, L3(D) is the volume of D, x is the length of the chord D ny, and integration is over T. (2) Formula (3.15.1) is suitable for the calculation of the measure of the set Ax = {(gu g2) e G x G : the point gx n g2 belongs to D}, where the domain D need not be convex. The result is dg, dg2 = 2nL2(D).

(6.7.3)

Similarly, integrating (3.15.2) we find that

L

de1de2=(n3/4)\\dD\\,

(6.7.4)

142

6 Basic integrals

where the set A2 is as follows: A2 = {(e, e2) e E x E: the line ex n e2 hits a convex domain Da

R3}.

The result (6.1.10) was used in (6.7.4), and \\dD\\ is the surface area of D. Integration of (3.15.3) over the set A3 = {(eu e2, e3) e E x E x E: the point e1ne2n

e3 lies in a domain D a R3}

yields

I

?! de 2 de3

= TT 4 L 3 (D).

(6.7.5)

The results (6.7.3)-(6.7.5) can be reinterpreted as special cases of more general relations which involve products of T-invariant measures in the spaces G or E. Let us look at this, restricting ourselves to measures in G x G. Let jubea T2-invariant measure on G. The image of the product measure ft x \i under the map is a T2-invariant measure in R2 and therefore coincides with c-L2, where L 2 is two-dimensional Lebesgue and c is some constant. This constant can be expressed in terms of the rose of directions m(d(p) and of the rose of hits X(cp) which correspond to \i (see §2.10), namely

-J

c =

X{(p)m{dcp).

Proof It is enough to find (/i x //) (A J (see (6.7.3)). For the moment let the line gx = (cp, p) be fixed and let x(di) be the total length of g1 n D. By the definition of k((p) the ju-measure of the lines g2 intersecting g^c^D equals k((p) • xidi)In accordance with fig. 6.3.1 we find

=

X((p)m(d(p)

x(9i)

= L2(D)-[*.{q>)m(dq>).

(6.7.6)

Because of (2.10.1) the constant c can also be written as a double integral: |sin(
- \l/)\m(dcp)m{d\j/).

Jo Jo

Let us restrict m to the class of probability measures on (0,ri).Then the above integral clearly remains bounded and it is natural to ask about its maximal value under this restriction. We show now that for every probability measure m on (0, n)

6.7 Integrals over product spaces

143

2 n and the equality holds only for the uniform m, i.e. in the case d<

(A \ = — ? . myacp) n This result is due to Davidson [59]. The proof we give now is due to Mecke. It admits generalizations to higher dimensions (see [53], [60], [61]). Since identically

f

sin \q> — ij/\dil/ = 2,

Jo m on (0, n) we have for every probability measure

2

sin \

n )o Jo ' where / stands for the uniform probability measure. Hence, writing (/ — m) for the corresponding signed measure

2

s i n |

and it remains to prove that the last double integral is non-positive. This can be done using a special integral representation for the function sin\q> — \\i|. Fig. 6.7.1 shows the graphs of the functions t = sin2 z

and

t = sin2(z — a)

(6.7.7)

and we have shaded the area under the graph of the function t = min[sin 2 z, sin2(z — a)']. The shaded area above the interval (0, n) equals r(n-a)/2 J-a/2

sin2 z dz = — — sin a,

(6.7.8)

144

6 Basic integrals

Because of periodicity, the shaded area above any interval of length n will have the same value. It follows that for every x, y e (0, n)

1

min[sin 2 (z — x), sin2(z — y)] dz = — — sin(|x — y\). o 2 The above identity can be rewritten as follows:

sin |x - y\ = J - f z

f * /(f)[0, sin2(z - x)]/(t)[0, sin2(z - y)] dr dz,

Jo Jo

where I{t)[a, b~] is the indicator function of the segment [a, &]. This is the desired representation for sin |x — y\. Integrating it with respect to the signed measure (/ — m) yields sin \x — y\-(l — m)(dx)(l

— m)(dy)

)o Jo

= - I I (/ - m)2{x : sin2(z - x) > t} dz dt, Jo Jo which is clearly non-positive. The uniqueness property of the uniform measure follows immediately from this. Our last remark refers to general M2-in variant measures on G x G. Let \i be such a measure which satisfies the additional condition //(the set of all parallel or antiparallel pairs of lines) = 0. A direct application of Haar factorization yields \x(dg2 dg2) = dM m(d\j/\

(6.7.9)

where we use the same notations as in §3.15.1, and m is some measure on the punctured circle SiXlthe angles \j/ •= 0 and ij/ = n). We stress that now // need not be a product measure, as was the case in (3.15.1). There exist measures m on the punctured circle for which dM m(d\j/) is not a measure on G x G in our usual sense since local finiteness can be violated. Using (6.5.5) we conclude easily that the condition

J

sin" 1 \l/m(d\l/) < oo

(6.7.10)

is both necessary and sufficient to have n({Qu QI) • 0i hits vl9 0 2 hits v2}) < oo for any two line segments vx and v2. It is not difficult to see that the set {(#i> 9i): both lines hit a square in IR2} can be covered by a finite number of sets of this type. Therefore (6.7.10) is a necessary and sufficient condition of local finiteness of the measures \i given by (6.7.9).

6.8 Kinematic analysis of realizations

145

6.8 Kinematic analysis of realizations The kinematic measure is a measure in the space of figures congruent to a given figure in Mn. It corresponds to (and often coincides with) the Haar measure on the Euclidean group Mn (see §3.13). Integration with respect to the kinematic measure plays a significant role in integral geometry, for instance in the Kinematische Hauptformel of Poincare-Blaschke; see [2] for a complete account. In this section we concentrate mainly on those results which we shall apply later on in the context of geometrical processes. I We start with the simplest versions of the Poincare-Blaschke formula. Let dd be the kinematic measure in the space Af of directed line segments 5 of fixed length \8\ (see §2.13). For a curve <£ a U2 and a line segment 3 we define n(S) to be the number of points of intersection of 5 with S£. Then n(8) dd = 4 -\5\ times the length of JSP.

(6.8.1)

Proof Since we have in this case d<5 = dM, the Haar measure on M29 our assertion follows from (3.13.3), (6.1.6) and the one-dimensional version of (6.6.3), that is from

n({g,t))dt = n(g)'\5\9 where n(g) is the number of points in the set g r\S£. Let Sf be a surface and let v be a directed line segment in IR3. We define n(v) to be the number of points of intersection of v with $f. Then n(v) dv = 4;r2 • | v| times the area of Sf,

(6.8.2)

where dv stands for the kinematic measure in the space Af (the proof is by (3.13.2), (6.1.8) and (6.6.3)). Let F be the space of circular flats of fixed area ||/|| in U3 (see §2.13,1V). A flat can be represented as where e is the plane carrying/and QeU3 determines the position of the center o f / o n the plane e. The kinematic measure d/has the representation (compare with (3.13.4)) where de is the y 3 -invariant measure in E and dQ is the planar Lebesgue measure.

6 Basic integrals

146

Let n(f) be the number of intersections of/ with a curve <£ in U3. By means of (6.1.7) and (6.6.3) we establish that n(f) d/ = 4TC2 11/11 times the length of <£.

(6.8.3)

II The previous results can be considered to be kinematic analogs of (6.6.3). There also exist kinematic analogs of the asymptotic results presented in §6.6. These refer to the cases where |<5|, |v| or ||/|| tend to zero. Let us consider the planar case. Our main interest will be in collections of linear segments rather than curves. Let m be a collection of segments on U2 of finite total length L. On Af we consider the function /k(<5) = Ik(3, m)=\

if the segment 8 has exactly k intersection points with m

= 0 otherwise. In contrast with (6.6.6), the function Ik(S) can be non-zero for arbitrarily small values of 15|, even for k ^ 2. For instance, if m. consists of segments situated as shown in fig. 6.8.1 then Ik(d) can equal 1 for all values of \3\ if k ^ 6. However, there can be significant differences in the order of magnitude of the integrals ak =

J

Ik(5)d5

as

|<5|->0

for the cases k = 1 and k > 1. Recall the well known result of Santalo [2] which gives the value of a 2 in the case m is a pair of rays (infinite in one direction) emerging from a node N. In this case

Figure 6.8.1 The segments 5l9..., S6 all lie in a halfplane bounded by the dotted line through N (the 'node').

6.8 Kinematic analysis of realizations

147

where 0 < \j/ < n is the angle between the rays. This result admits a number of successive generalizations. If m consists of more than two infinite rays emerging from a node then ak = Q-|<5|2.

(6.8.5)

The quadratic dependence on |<5| follows from purely qualitative considerations. To see this we represent the kinematic measure in the form d<5 = AQ dcp,

where d<2 is planar Lebesgue and dcp is the angular measure on S x . For each value of cp, integration with respect to dQ gives the area of a domain in U3. By homothety this area equals ck(cp)\5\2, where ck(cp) does not depend on |<5|. Clearly then

•J Also, the values of C can be found using (6.8.4) and applying the inclusionck(cp) dcp.

k

exclusion rule. We leave the details to the reader who may consult [42]. The sequence C2, C 3 , . . . is essentially finite; after a certain index k all Q's are zero. The index of the last Ck which is non-zero can be called the order of a node. Nodes of the order two are shown in fig. 6.8.2; they appear in a stochastic context in chapter 7. Equation (6.8.5) are no longer valid for all values of |<5| if the rays emerging from a node are of finite length. Yet they remain valid for sufficiently small values of | <5 | (where the truncation of the rays does not affect the sets {(5:/fc(<5)= 1}

forfc = 2 , 3 , . . . .

From this remark and from (6.8.1) follows an asymptotic result valid for every *n (a finite union of line segments of finite total length L), namely iffc= 1 = |<5|2-£ Q(JVi) + o(|<5|2)

TV angle

knot

cross

Figure 6.8.2 Nodes of order two

if* > 1,

(6.8.6)

148

6 Basic integrals

where the points Nt are the nodes of m (i.e. points where at least two linear segments meet at non-zero angles). Ill What will happen to (6.8.6) if we allow m to possess curvilinear parts? The answer is that under appropriate smoothness conditions the nature of (6.8.6) will not change as far as in the expansion only the terms of order |<5| and |£| 2 are considered. We explain this using an example. Let m consist of two intersecting circles m^ and ^ 2 ( s e e fig- 6.8.3). Then we have I2(5, m) = Jn(<5, m) + I22(S, m) + I12(S, *n\ where Ia(S, m) — 1 = 0

if both intersections are with ^ni otherwise,

and J12(<5, m)—\ = 0

if one intersection is with mx and the other with <m2 otherwise.

In clear notation

f/«(a, ~) dd ^ !i2(s9 mt) dd = | - 1 ^ where r£ is the radius of the circle <mt. (In fact the latter integral can be calculated explicitly using (3.13.3)). On the other hand,

where C2(N1) can be calculated using (6.8.4):

Figure 6.8.3

6.8 Kinematic analysis of realizations

149

(the angle \j/ is shown in fig. 6.8.3). Here we have used the following general principle. If the linear rays emerging from a node N are replaced by smooth curvilinear arcs which emerge from N in the same directions and which do not create new nodes, then (6.8.5) is replaced by with the same values of Ck. The proof can be by integration of (3.13.3) over the corresponding difference sets. From the above remarks (6.8.6) follows for k = 2. For k = 3, 4 we have and therefore Ik(d,*n)dd = o(\d\2l

/c = 3,4,

which also conforms with (6.8.6). IV Equation (6.8.6) admits a generalization to the case where the participating set m need not be bounded. We will assume below that the set <m n b(r, O) (as usual b(r9 O) is the disc of radius r centered at O) can be represented as a union of a finite number of line segments for every r > 0. In other words, we assume that m is a realization of a segment process, see §7.13. Let us choose r in such a way that there be no nodes of m on db(r, O) (the aim is to avoid certain 'boundary effects' which make the things more complicated). The desired modification of (6.8.6) is as follows: lim 1*1-0 2

lim \3\~

I

I5I" 1

/1(<5, Jb(r,O)

Ik(S9 m)dd

=

(6.8.7) X

Ck(Nt)

if

k > 2,

where b(r, O) = {6 e Af; the origin of the directed segment 3 belongs to b(r, 0)}, L(r) = the total length of m n b(r9 O). Similar equations also hold for the sets m consisting of smooth curvilinear arcs (see the remarks of subsection III). V Let m be from the class described in IV. Together with the number of points in 3 n m we can consider the angles at which the intersections occur. The

150

6 Basic integrals

Figure 6.8.4. The segments with no arrow on them belong to <m

situation is simplest if we have only one point in 3 n ^ , i.e. if 3 belongs to the set Bx = {3eA%:I1(3,™)=

1}.

On B1 the intersection angle \\i = il/(3) is a well-defined function, see fig. 6.8.4. Let fi be an arc (or, more generally, a Borel set) from § x and let Ip(\jj) = 1 = 0

if \jj belongs to p otherwise.

Then

\im\3\~1

I

Jkr,O)

1,(3, *i)IfiW) d<5 = L(r) I I sin iA | # . Jfi

J

(6.8.8)

Proof We split the set b(r, O) n Bx into non-intersecting subsets so that in different subsets 3 hits different segments from m. Let us denote the latter segments by v l 5 . . . , vs. For every 3 e b(r, O) excluding a set whose measure is we have

I1(89m)=£l1(8,vi). For |^| ^ 0

f JHr,O)

h(S, vdW) d3 = f I,(S, vtt n b(r9 O))IfiW) d3 + o(\3\). JAJ

As follows from (3.13.3) and (3.7.3)

f/i(5, v)W) d^ = |(5| • |v| • f for every value of |<5| and for every segment v a U2. Equation (6.6.8) follows from these observations. A set of segments on U2 is called a mosaic if it induces a partition of IR2 into bounded convex polygons whose interiors do not pairwise intersect but the union of their closures gives the whole plane; the number of nodes in any disc b(r, 0) should be finite. A mosaic, m> cannot possess nodes of angle type, while other types shown in fig. 6.8.2 are not excluded. Let m be a mosaic possessing only nodes of order two. Then on the set

6.8 Kinematic analysis of realizations

151

Figure 6.8.5. 3 hits two segments from a mosaic which meet at a node

B2 = {<5 e AJ : J2(<5, **) = 1} we define the functions (angles) ij/^3) and \j/2{d) as shown in fig. 6.8.5. We choose r in such a way that there are no nodes of <m on db(r, O) and consider the sets Btj = {S e b(r, 0): S hits two segments vi9 vj e <m which meet at a node}. If | (51 is sufficiently small then

B2nb(r90) = [JBip the sets in the above union are pairwise disjoint. For such values of |<5|

\n-il/1 -\j/2\ = Oy, where O 0 is a constant on Btj and equals the angle between the segments vt and Vj. By means of (6.8.4) we find , 1 + (n — | n — if/i — \jj21) c o t 17i — ij/1 — \\i2

Summation over all sets Btj yields l i m \5\~

- \n - i//1 - i//2\) c o t |TT - ^

-

= 3nf r,O)

1 + (7T -

= 2n(nf + nc) + n- nk,

|7T - ^

- ^ 2 | ) COt

|TT

- fa -

fa\

(6.8.9)

where nf, nc and nk are the numbers of nodes of fork, cross and knot types, respectively within b(r, 0). We use (6.8.9) in §7.14. VI Now let *n be a collection of countably many lines gt e G. We will assume that the sequence gt has no accunulation points in G. In other words, m, can be viewed as a realization of a line process in the plane (see §7.5).

152

6 Basic integrals

O

Figure 6.8.6 The distance between Vj and v2 is d

There is a counterpart of (6.6.9) for realizations of line processes which we are now going to derive using the symbolism of the Haar measure on the group fV02 rather than that of the kinematic measure on Af (they are equivalent). First we consider two infinitesimal segments vl9 v2 situated as shown in fig. 6.8.6. We write IB(m) = 1 = 0

if a line from m hits both vx and v2 otherwise.

We consider also several fixed line segments 5l,..., Ss in U2 and let kx,..., be non-negative integers. We write lA{m) = 1 = 0

ks

if exactly kt lines from m, hit di9 i = 1,..., s otherwise.

As usual, by M<m we denote the result of applying a motion M to m. For a fixed m9 IB(M*n) and lA(Mm) are functions defined o n M 2 . We consider the asymptotic (as / = \vx\ = |v2| -• 0) behavior of the integral )IA(M*n) dM, JM )MOeb(r,O)

where dM is the Haar measure on M 2 . Because in m there are only finitely many lines hitting any b(r9 0) for smaller values of |vf|, the set {M: IB(M&t) = 1 } can be split into pairwise non-intersecting subsets of the type Ct = {M : the line from the set M<m which hits both vx and v2 is the image (under M) of some fixed line gt e m}. By an easy modification of the argument of §6.4 we find that IA(Mm) dM = I2 d'1 I

UA(Mit + ttm) + IA(Mu-ttm)']

dt + o(l2).

It is necessary to explain Mit + tt (M" 1 is the inverse of M). We consider two oppositely directed segments q± and q2 both of unit length, both lying on gt and having a common origin at tex(gi) = ginb(r90). By definition, Mu + t corresponds to qx to Mit _ it corresponds to q2 (in the sense of §2.13). The measure dt is one-dimensional Lebesgue on gt.

6.9 Pleijel identity

153

By summation we get lim r2d

f

I

In(M*n)lA{M*n) AM

= E ff ( )

M.\ + ,i"0 + h(M~u-,M~\ &t.

(6.8.10)

x 9i

9i hits b(r,O) J x(9i)

We stress that here again the value of the limit does not depend on the shifts of the pair vx and v2 along the x-axis. We use this result in §10.1.

6.9 Pleijel identity A family of identities primarily associated with isoperimetric inequalities for planar convex domains was discovered by Pleijel [37], [38] in 1956. It has proved to be closely related to the questions discussed in our chapter 5. In fact below we derive them by applying a simple integration procedure to (5.4.1). It was shown in [3] that by integration of other combinatorial formulae various generalizations of the Pleijel identities can be derived in a rather systematic way. Generalizations can be extended to more dimensions as well as to non-convex domains. Some of them can be applied in stochastic geometry, as was actually done in [3]. In this section we treat only the simplest case. Let D be a bounded convex domain in IR2, and let # l 5 ...,#„ be w linear chords of D. We will imagine these chords to be directed, therefore one endpoint of each Xi will be called the 'head' and the other the 'tail'. In the space of sequences (xl9..., Xn) w e consider the product measure dfiin) = dx1...dxn, where each d/£ corresponds to the M 2 -invariant measure on G. First we assume that the boundary dD does not possess linear parts. Under this assumption the set of endpoints of our chords will possess triplets lying on a line for a set of sequences of measure zero. Therefore (5.4.1) (which we now consider as written for dt = xd holds 'almost everywhere' and we can integrate it with respect to the measure fi{n). We will use the notation \x\ for the length of a chord, considered either as a function defined on {g e G : g hits D} or as a function depending on an ordered pair of points from dD. Several times in the integration we will use the identity

where g e G is fixed and

154

6 Basic integrals

Integration of the left-hand side of (5.4.1) yields

f We turn now to the right-hand side of (5.4.1). Using symmetry we find that

2 f £ 1^14-ite) d/i(w) = In f |xi|/B-i(Zi) d/^> ^ JG

JG

Again by symmetry

= 2n(n

•J'

In this expression Xu denotes the line segment joining the tails of Xi and X2, and Id and / s are the indicators of the events Xn e {du} and Xn e {sfc}> respectively. After collecting similar terms and dividing by (n — 1) the preliminary result of integration of (5.4.1) is written as

- f (4|zir dg = In [I

X^UAXn)

~ /.(Zi 2 )] d^(M). (6.9.1)

Using (3.7.3) for dg, the right-hand side of (6.9.1) is written as n 1A1 sin ^2 diAi d^ 2 d/x d/2,

JG G xG

and here integrations over \//1 and ^ 2

can

be performed. For fixed /1? l2 6

sin ^ x sir sin il/2Ud(x) — h(x)^\ d*Ai d\j/2 = — 4 cos ccx cos a 2 ,

Jo Jo where the angles ax and a 2 are shown in fig. 6.9.1. Thus the final result of integration is the Pleijel identity for convex domains:

Figure 6.9.1 The angles ax and a 2 lie in one halfplane with respect to the inside of D

6.9 Pleijel identity

155

l" dg = U- f f Ixr1 cos a i cos a2 d/x d/2.

(6.9.2)

During derivation we have assumed that n > 1, but in fact this restriction is not significant. Making use of the linearity of (6.9.2) we establish A\X\)

dg = -

f'{\x\) cos a i cos a2 d/x d/2

2

G

(6.9.3)

JJ

firstly for polynomial functions / of the form b2x2 + ••• + bnxn (where / ' denotes the derivative). Then, by Weierstrass's approximation theorem, (6.9.3) holds for every function / having continuous derivative / ' and satisfying /(0) = 0. In particular if we take f(x) = x the integral on the left-hand side reduces to n- ||D||, where ||D|| is the area of D (see (6.3.1)). On the other hand, we can use (3.7.4) to get fdxDlxl'1

/(Izl) dg = -

sin <xx sin a 2 d/x d/2.

2

(dD)

Thus, putting f(x) = xwe get two equations: 1 CC n- \\D\\ = cos d1 cos a2 d/x d/2 (dD)2

and 1 CC n\\D\\ = sin ax sin a 2 d/x d/2 (dD)2

from which we get, by addition,

II

1

*• AI

AI

or in a weaker form which is the classical isoperimetric inequality. In §6.10 we will need the following corollary of (6.9.3). Using (3.7.4) we rewrite this equation in the form

f /(lxl)dflf= f A l z l ) l z | c o t a i c o t a 2 d ^ . JG

(6.9.4)

JG

The above transformation is legitimate only for planar convex domains D whose boundaries do not possess line segments. If this is not the case, additional terms appear. Consider for example a polygonal domain D with sides at of length \at\. In this case, it is necessary to add the integrals

156

6 Basic integrals

where p is the distance between /x and / 2 . By simple calculation we get the following modification of (6.9.4), valid for bounded convex polygons D:

f f(\x\)dg = f ff(\x\)'\x\cot0L1cot0L2dg + Yt (*' f(u)du. (6.9.5) JG

JG

JO

Note that in (6.9.5) the condition /(0) = 0 is no longer necessary.

6.10 Chords through convex polygons In this section we assume that D is a polygon. If in (6.9.5) we formally put fy(u) = 0 if u
tfu>y,

(6.10.1)

then the left-hand integral there will equal ii{geG:\x(g)\>y}; i.e. the invariant measure of the set of chords of D whose length exceeds y. We note that the distribution function of the length of a random chord x through D is usually defined to be F(y)=l-\dD\-1/jL{gsG:\x(g)\>y}9 where \dD\ is the length of the perimeter of D. The derivative of fy(u) as given in (6.10.1) should be replaced by Dirac's ^-function concentrated at y. Thus (6.9.5) becomes \\-F(yj\-\dD\

= 1 \\

(6.10.2)

[a,] n [<*_,]

where [ a j n [a 7 ] is the set of lines hitting both sides at and a, of D, and x + = x if x > 0, 0 otherwise. In each of the double integrals one integration can be performed by passing to the \x\9

Figure 6,10.1 (p is the direction of g

6.10 Chords through convex polygons

157

Figure 6.10.2

dg =

sin <x1 sin a 2

sin ((x1 + a 2 )

d\x\'d
Clearly we can put oc1 = (p,OL2 = n — (0ii + a j ,

where d{j is the angle between at and a,- (see fig. 6.10.2). Thus we have JJ

[a,] n [aj]

S (\x\) cot ax cot a 2 dg = —r—— cos cp cos(0« + cp) sin ^ J*ij(y) (6.10.3)

where the domain of integration is ®ij(y) — {*A : a chord joining at and aj exists which has direction cp and its length is y). The reader will check without difficulty that if the sides ai9 aj have a common vertex, then the set
1

sin(0t-; + 2cp,).

The endpoints of the intervals in question are elementary functions depending on Oij, \at\ and |oy|. This means that if the sides at and o^ have a common vertex then the integral (6.10.3) can be expressed elementarily in terms of these quantities. The above restriction on the choice of a{ and a, can be removed by virtue of the indicator functions relations the intervals bt and bj are shown in fig. 6.10.2. We conclude that a similar principle is valid for every integral appearing in (6.10.2). Thus the integral (6.10.3) can be expressed as an elementary function X(ah a^ of the segments at and aj.

6 Basic integrals

158

Clearly the above yields a convenient algorithm for calculation of random chord length distribution for convex polygons (with no pairs of sides parallel): y)+.

[1 - F(y)l • \dD\ = X X(ai9 a,) + £ (\at\ -

6.11 Integral functions for measures in the space of triangular shapes In this section we describe some results referring to measures in the space L 2 whose densities were derived in §3.16, III. We give the values of these measures on the family of sets A(x) = {triangular shapes for which the minimal interior angle is greater than x} (see Fig. 6.11.1). More elaborate expressions for families of sets depending on more parameters have been obtained by Sukiasian in [39] and by Oganian in [40]. We start with the measure vs which has a density given by (3.16.7). We have f*n — 2x

vs(A(x)) = 2

f*n—a—x

da

[sin a sin P sin(a + jS)]"1 dp.

Making use of the identity sin a [sin P sin(a + jS)]"1 = cot P - cot(a + p\ we integrate: rn-2x

vs(A(x )) = 2

sin" 2 a da

Jx

-A

rn-*-x

[cot p - cot(a + Jx

[In sin (a + x) — In sin x] sin

2

a da

-2x

ln[sin(a + x) sin" 1 x] d(cot a) rn-2x

= 4 cot x ln(2 cos x) + 4

cot a cot(a + x) da.

A(x)

Figure 6.11.1 The two components of A(x) are triangles symmetrical with respect to the diagonal

6.11 Integral functions for measures in the space of triangular shapes

159

Since cot a • cot(a + x) = cot x(cot a - cot(a + x)) — 1 we get

r %

n-2x

cot a cot(a + x) da = 3x — n + 2 cot x ln(2 cos x).

Thus finally for x e (0, TT/3) vs(A(x)) = 4(3x - 7c + 3 cot x ln(2 cos x)).

(6.11.1)

Note that this expression implies lim vs(^(x)) = 0 X-+K/3

(which can be considered as a check of (6.11.1)) and lim vs(^(x)) = oo in concordance with (3.17.3). Let us now consider the density (3.16.6). In terms of the quantities

this density can be rewritten as ^ z ( l + z2)2u(l + w2)2(l - zu)(u + z)" 2 . Therefore the problem of calculation of vh(A(x)) reduces to integration of this rational function. The result will be vh(A(x)) = ^ - y - (^

jj^

+ ^b5 +

+ 159b5 + 2% 3 - 31b),

where b = tg x/2. The above is valid for x e (0,rc/3);for x x = 0 we get

G (TC/3,

7r) clearly vft(>l(x)) = 0. Putting

which was earlier found by Kendall in [41]. The values mh(A(x)) can be found by starting from the observation that the density (3.16.3) equals the expression ^ . ai . a 2 a, + a2 2 sin — sin — cos which is very convenient for integration. The result has the form

160

6 Basic integrals

mh(A(x)) = 3 cos x + 3 cos 2x — 3(n — 3x) sin x for x G (0, n/3) and zero for x (n/3, ft). Putting x = 0 for the total measure we get the value (3.17.1). Similar integration for the density (3.16.10) is even more simple. We present the result: for x e (0, n/3) vA(A(x)) = —2 (ft — 3x) cos 2x + —2 (sin 4x + sin 2x). Hence (3.17.2).

Stochastic point processes

Random point processes in different spaces will be the main object of our study for the rest of the book. We start this chapter with a brief exposition of basic notions. The measure-theoretic ideas used here are the same for all 'carrier' spaces that we consider. However, the clearest geometrical interpretations can be given for point processes on the line, which we treat in more detail to illustrate our ideas. We begin the study of point processes that are invariant with respect to groups with several concrete examples. Emphasis is put on models which reappear in further chapters. We also introduce the important notions of marked point processes and moment measures. The 'analysis of realizations' of the previous chapter is put to use by means of an averaging procedure. In particular, results concerning the nodes in segment processes and random mosaics in U2 are obtained by this method. It can be broadly used in the study of invariant geometrical processes, and further instances of this are scattered throughout the remaining chapters.

7.1 Point processes The notion of a point process is a formalization of the intuitive idea of a random set with countably many points. A precise definition is as follows. Let X be a space (for instance a manifold or, more generally, a complete separable metric space, see [18]) which is to 'carry' the realizations of the point process. A realization <m a X is a subset which has no condensation points in X. We denote by Ji the class of all realizations, m e M. Where necessary we will use the notation M = Jtx which stresses the basic space X.

162

7 Stochastic point processes

Remark By this definition realizations cannot possess multiple points, i.e. we consider only 'simple' point processes in the terminology of [18]. Let B c X be Borel. We put N(B, m) = card Bn<m. Let si be the minimal a-algebra of subsets of Jt which renders all the functions N(B, m) measurable. Equivalently, si is the a-algebra in M generated by sets of the type {^ei:

N(B, m) = k).

A point process in X is a measurable map **(©): ft-> Uir, (7.1.1) where (H, J*\ p) is a probability space. The above means that to any element co G H corresponds a realization **&(©>), with the only requirement that { « : ^((o) e A} e 3F

as soon as A belongs to si. (Some examples of actual maps m((o) can be found in §§7.8-7.10.) Every point process m((o) induces a probability P on (M, si) by the formula: P(A) = p{co : m(&) eA}, Ae s/. This P is called the distribution of m(
& = $4,

p= p

and define <m((o) by the identity map wi —• m.

(7.1.2)

Probabilities on (M, stf) can be considered in their own right, apart from the notion of a point process. In such cases they are usually described by means of the theorems of continuation of probabilities (measures), starting from values of P given on some subclass sf0 a si. Thus arises the general problem of the description of the classes s/0 from which continuation is possible in a unique way. Still another aspect of the same question is to describe functions which are defined on si0 and which permit continuation to a probability on (Ji, J / ) . Below we give the answers to these two questions for a number of spaces X using simple geometrical considerations, which are clearest for X = R, which we treat in some detail. Other cases can be treated similarly.

7.2 A-subsets of a linear interval Let [a, b) be a finite semiopen interval on U: [a9b) =

{xeU:a^x
Here and in §7.3 and §7.4, we will consider only similarly defined semiopen intervals to be termed simply as 'intervals'.

7.2 /c-subsets of a linear interval

163

Figure 7.2.1 a = 0, b = 1

We will call the subsets of [a, b) which contain k points k-subsets. The space of all /c-subsets will be denoted by [a, b)k (in contrast to la, b)k = la, b) x • • • x [a, b), k times). An element of [a, b)k is written as m= {x l 9 ...,X k } (as usual, the brackets { } denote sets, while the brackets ( ) denote ordered sequences). There is only one zero-subset of [a, b), namely the empty set. Thus the space la, b)0 consists of one element (is non-empty!) Also it is clear that la, fc)i = la, b).

To construct [a, b)2 we use the map

{xl9x2}^(£l,£2), where ^ = min x l 9 x2,

£2 = max x l 9 x2.

This map identifies [a, b)2 with the triangle S2 (see fig. 7.2.1). The diagonal does not belong to S2. In general, the 'ordering' map where fi = m i n x l 5 . . . , x f c ,

...,

£k = max x l 9 . . . , xk,

k

maps [a, b)k onto a simplex Sk in IR , which in terms of coordinates £ l 5 . . . , £k is described by the inequalities

a ^ ^ < ••• < £k < fe. The above maps reduce the task of the construction of measures on [a,fr)fcto the construction of measures on (subsets of) Euclidean spaces. In particular we can use usual rectangles in Uk, noting that rectangles which lie completely within Sk make up a semiring of subsets of Sk and they generate the Borel subsets of Sk. We denote this semiring by J^k(a, b). A rectangle r e J^k(a, b) has the following interpretation in terms of /c-sets. Given r e #£k{a, b), we consider the projections /i,...,4 of r on the coordinate axis ^ , . . . , £fc, so that

(7-2.1)

7 Stochastic point processes

164

Figure 7.2.2 r = Ix x ••• x Ik.

The sequence (7.2.1) satisfies the following conditions: (a) Il9...,Ik are pairwise non-intersecting and belong to [a, ft); (b) Is lies to the right of It if 5 > /. In fact (a) and (b) are also sufficient: if both (a) and (b) are satisfied then the product Ix x • • • x Ik falls within Sk (see fig. 7.2.2). By the construction of the ordering map we have r = {<m e [a, b)k: card mC\It = 1, i = 1,..., fc}. Below we will use the following notation. Let Bl,..., Bs be a system of Borel sets from [a, b\ and let ll9..., /s be non-negative integers. We write /

Bs

= {<m G [a, b)k: card

= li9 i = 1 , . . . , 5}.

In this notation r =

We have also seen that *, b) =

h

h

r

" 1

Ik\

intervals

Il9...,Ik

satisfy (a) and (b)

By applying the well known facts of measure theory (see [46]) we come to the following conclusions. Every measure on [a, b)k is completely defined by its values on the events

where the intervals satisfy (a) and (b) (i.e. on the events from Jfk(a, /?)). A non-negative function F defined on J^k(a, b) is a measure on [a, b)k iff it is countably additive within the same class of events. Example Let F be defined on Jfk(a9 b) as follows: (7.2.2)

7.3 Finite sets on la, b)

165

Figure 7.2.3

where m is some atomless measure on [a, b). Clearly F gives the values of the product measure m x • • • x m (k times) on rectangles in Sk, and therefore coincides with restriction of the mentioned product measure to Sk, i.e. F = m x •••x m

(k times)

on Sk.

Clearly F(Sk) = (m\ F

;:--t»=n

provided 7X, ..., 7S c [a, ticular, if k = 2, the event

(7.2.3)

pairwise do not intersect, and

ff = /c. In par-

can be identified with a triangle in S2 (see fig.

7.2.3) and by the last formula its m x m measure equals \(m(I))2. In the case in which m([a, b)) < oo, the measure F given by (1.2.2) can be normalized to yield a probability measure We will have (under the same conditions as in (7.2.3)) m(It) We conclude that the random set described by the above probability can be obtained by an 'experiment' in which k independent random points are dropped on [a, b\ each point having probability distribution proportional to m.

7.3 Finite sets on [a, b) We consider the space of all subsets of [a, b) which possess only a finite number of points, i.e. (see §7.1) ^ia,b) = {m ^ [a> b): card m < oo}. Clearly this space can be represented as a union: (7.3.1) fc=O

166

7 Stochastic point processes

)

f

7

)

Figure 7.3.1 We will use the notation 1 , . . . , s \ = {m 6 Jf{a,b) • card m n It = li9 i = 1 , . . . , s}9 L'i **J where / x , . . . , / s are intervals from [a, b), and / x ,..., ls are non-negative integers. We denote by Jf(a,fe)the class of subsets oiMlah) of the above type, which we define by the conditions

(a) lx,..., Is are disjoint and their union is [a, b)\ (b) each /, is either 0 or 1. Let us show that J f (a, b)=\J

Mf&a, b).

(7.3.2)

fc = O

Indeed, if A e J f (a, b) then each element ^ e ^ l has a fixed number of points, namely k = lx + • • • + /s, and therefore A c [a, b)k. Conversely, if

then we can represent this A in the form

'"• o where Jl9..., Jjis the collection of disjoint intervals whose union is ((J *=1 /j)c (c stands for complement, see fig. 7.3.1). Because of (7.3.1) every measure on Jf{a,b) *s completely determined by its restrictions to the sets [a, b)k, /c = 0, 1 , . . . . Therefore (see §7.2) we obtain the following proposition. Let F be a non-negative function defined on Jf(a, b) (and therefore on each (a, b)). If on every 3#l(a, b) the function F is tx-additive then F is (equivalently, can be extended to) a measure on Ji^hy We are especially interested in the case in which because we can put

P = c~1F

and obtain a probability measure on Jt[aJj). Example Let F be defined on J f (a, b) in the following way:

7.3 Finite sets on [a, b)

167

where m is a /fm'te atomless measure on [a, b). The restrictions of F on every J^(a,ft)are essentially the measures which we have considered in the Example in §7.2. Therefore F in (7.3.3) is a measure on J(laM generated by the measures m x • • • x m (k times) on each [a, b)k. In particular

The corresponding probability therefore is P = exp( —m([a, b)))-F.

(7.3.4)

Let Ix,..., / s be an arbitrary system of pairwise non-intersecting intervals on [a, b). For every probability P on Mla b) we have

!•([•;• VL'i

;•!)= i p h , ' - , J ' h\J

r,trj

VL'i

I* rt

J

'l), rjjj

where Jt,..., Jj are non-intersecting intervals which complement (J /, to [a, b). The event

h belongs to [a,fc)/?where / = £ ' * + £ r,-. By (7.2.3) for P defined by (7.3.3) and (7.3.4) the probability of this event equals

n ^ r t f

(7.3.5)

(we used additivity of m). Performing the summation we find that

The probability measure P = Pm defined by (7.3.3) and (7.3.4) (or, equivalently, by (7.3.6)) will be termed Poisson, governed by the measure m. Equation (7.3.6) now can be expressed in words as follows. For any Poisson probability on Jt[ayb) the events

for non-overlapping

intervals are independent, the distributions

are usual Poisson with parameter m(I). We mention also the following property. The conditional distribution of a Poisson point process on [a, b) described by (7.3.6) conditional upon the event

168

7 Stochastic point processes

is proportional to the fc-fold product of the measure m with itself. The proof is a direct corollary of our construction.

7.4 Consistent families Let [c, d) be a subinterval of [a, b). The realizations space Ji[a mapped onto Ji^cd) by means of the truncation *n-+ *nC\\_C, d\

b)

can be (7.4.1)

where m e J?[a,by Assume that two probability measures, Px on Mlah) and P2 on J£[Ctd), are given. We say that Px and P2 are consistent if P2 is the image of P^^ under the map (7.4.1). A check that Px and P2 are consistent can be reduced to the verification that p2 (A) = Px {A)

for every A e jf(c9 d).

(7.4.2)

Now let m((o) be a point process on the line (see §7.1, X = IR), and let P be the distribution of m(
and we denote by P[aM the distribution of the latter. The family of distributions {P[a,b)} which arises in this way is consistent. This means that for each [c, d) and [a, b) such that [c, d~] a [a, b) the probabilities P[ab) and P[cd) are consistent. The last assertion follows from the relation (m(
The proof is by standard measure-theoretic methods and is therefore omitted. Example 1 Let m be an atomless measure on IR. Let P[ab) be Poisson governed by the restriction of m to the interval [a, b). By (7.3.6) this family of probabilities is consistent. We conclude that this family is generated by a probability P = Pm on Jtu. This Pm we again call Poisson, governed by m. Any point process on IR with distribution Pm (which exists by (7.1.2)) is also Poisson, governed by m.

7.4 Consistent families

169

Figure 7.4.1

The above theory yields the following answer to the queries mentioned in §7.1. Corollary Let P be a probability on JKU. The values of P on the class of events JT = U *(a, b), where the union extends over all intervals [a, ft), define P uniquely. The proof follows from the uniqueness statements of this and previous sections. We emphasize that the class J f consists of elements of the form ,...,

. = {m G Jfu : card m n It = lh i = 1,..., s},

where / x , ..., / s are pairwise disjoint intervals on R with union again an interval (see fig. 7.4.1) and each lt is either 0 or 1. With the preceding theory we can state some conditions which guarantee that a non-negative function F defined on ^f is a probability measure on Jtu. They are as follows: (a) the restriction of F on every J^k(a, b) should be a measure; (b) for every [a, b) we should have

i.e. on every Jtlah) the function F generates a probability measure; (c) the family of probability measures mentioned in (b) should be consistent. For the values of F on concrete sets we use the notation

and similar slight abuse of notation will persist in other parts of the book. A rather special but important way of constructing functionals F which satisfy (a), (b) and (c) is by means of so-called relative density functions (r.d.f.-s). We consider non-negative functions fI(x1,..., xn) which depend on an interval / c: R as well as on points xl9..., xn s I and which are symmetrical in the arguments xt. In the case n = 0, / 7 depends merely on /. Given a system of functions

170

7 Stochastic point processes

we construct an F as follows:

In this expression hi = ' " = hn = 1»

a

H other /fc are zero,

Clearly (7.4.3) implies (a) ( / / ( x 1 ? . . . , xk) happens to be the density of the measure in question), (b) follows from the condition

(b')£ (n!)"1 f-f/ / (x 1 ,...,x ll n

n=o J i J Together with the consistency condition (c) this guarantees the existence of a probability P on J?u (or of a point process on U). We refer to this probability as generated by the system {//(x 1? ..., xn)}n=0fl 2 , the latter functions are called the corresponding r.d.f.-s. The term r.d.f. is due to the interpretation /dx,

r i

dx n / \ ( U dx ; )\ _

, . . . , ,

i — j / i X j , . . . , xnj a x x . . . a x n

which follows immediately from (7.4.3) if we take there F = P, the intervals Iil9 ..., Iin we assume centered at x l 5 . . . , xn and having infinitesimal lengths d x 1 ? . . . , dx n . Here the assumption of continuity is crucial. Example 2 Let m be some measure on U possessing a density p(x). We put // = e~m(/)

/7(xl5...,xJ = np(^" m(/) . Because

(7.4.4)

J>

(a) and (b') are satisfied. Moreover, we identify F generated by this system via (7.4.3) with F of the previous example. We conclude that (7.4.4) is a system of r.d.f.-s and it generates the Poisson Pm. If continuous r.d.f.-s are subject to the condition fI(xl9...,xH)^Cn

(7.4.5)

for some constant C then necessarily ...,xn)dx1...dxn. The functions cp are called the densities of P. We have

(7.4.6)

7.5 Situation in other spaces

171

(7.4.7)

where l{ is the interval which separates dxt from dx i + 1 . The condition (7.4.5) enables us to apply the dominated convergence theorem and interchange in (7.4.7) the order of limiting and summation operations. The existence of the limits of the summands will follow from the expressions of the corresponding probabilities by means of integrals which generalize (7.4.3). The r.d.f.-s can be expressed in terms of the densities as follows

(-IT m!

• • •
dum

,8) (7.4.8

This expression can be viewed as an integro-differential counterpart of the inclusion-exclusion type formula valid for an arbitrary set of events {Ai}fil:

(the last summation is over subsequences Jk c {m + 1,..., M} of size k). Connections with (7.4.8) become clear when we put

for an appropriately chosen set Sx,..., SM of infinitesimal (M -• oo) intervals for which 8t ndj = 0 if i / j , [j Si = I. There is an important question: what are the conditions which guarantee that a system of non-negative symmetrical functions

(p(xu...,xn)9

n = 1, 2,...

actually generates a probability P via (7.4.8) and (7.4.3)? It is not difficult to prove (see [64]) that (1) the existence of a constant C such that
n=l929...

and

(7.4.9)

(2) the right-hand side of (7.4.8) is non-negative for every / and n, are sufficient conditions.

7.5 Situation in other spaces The procedure of construction of probabilities on (M9 s/) outlined in §§7.2-7.4 for the case X = U can be used with minimal alterations for other spaces X if

172

7 Stochastic point processes

an appropriate class of subsets of X with which to replace the linear intervals is at hand. The basic and only properties of the intervals on IR which we used in the above construction are that they constitute a semiring which generates the Borel sets of IR. Therefore, the class of subsets of X in question should also be a semiring generating the Borel sets in X. If such a semiring S can be identified, then all the statements of §§7.2-7.4 can be transferred to point processes in X by simply replacing the intervals in their formulation by elements of S. Recall that S is called a semiring if (see [46]) (1) 5 is closed with respect to taking finite intersection, and, (2) if A, B e S, then the relative complement of A in B can be represented as a union of finite number of pairwise non-intersecting elements from S. In the spaces

_ X = R", G, M 2 ( = Af), A2 (7.5.1) and others there are natural semirings of subsets. Let us describe them for the spaces in (7.5.1) (since we later consider point processes in these spaces). These spaces have certain product representations (see chapters 2-4), therefore the following well-known result will be useful. Let X t and X 2 be two spaces and let St be a semiring in Xt which generates the Borel sets of Xi9 i = 1, 2. Then the product sets Ax x A2 with Ax e Sl9 A2e S2 constitute a semiring which generates Borel sets in X1 x X 2 . The factor spaces which we need are the line IR and the circle S x . On IR we use the intervals; on S x we use the arcs (under appropriate conventions concerning the boundaries). We obtain table 7.5.1 of semirings for spaces in (7.5.1). It is clear from the remarks above that to specify a probability on the spaces from (7.5.1), say, requires knowledge of the probabilities of the events of the type

Table 7.5.1 The space X

Product representation ofX

oM. n

IR x U x • • • U (n times)

G

Sj x R

y2

§ ! X U2

A2

U2 x (0, oo) x U x S j

Elements of S 7X x • • • x /„, where each /, is a linear interval a x / (a 'shield'), where a is an arc and / is a linear interval 7X x 12 x a, where /,, a are as above /i x I2 x J x 73 x a, where Ih a are as above, and J a (0, oo) is an interval

7.6 The example of L. Shepp

A

l

A\ ' *' *' / S

=

{"* e ^X

:card

173

^ ^ i44 = /i9 i = 1 , . . . , s}9

where / 1 ? ..., /s are non-negative integers. In fact it is sufficient to consider the class J^x of such events defined by the following conditions: (a) each Ai belongs to S; (b) At nAj= 0 if i ^j and (J 4 £ e S; (c) each Zf is either 0 or 1. (Compare with §7.4.) We can also consider Poisson probabilities (and therefore Poisson point processes) on these spaces governed by measures. The basic formula here is the same as (7.3.6); for events from J^x w e have

where m is a measure on X. Using the extension of probability, it is possible to show that (7.5.2) remains valid if At a X are arbitrary Borel and lt are arbitrary non-negative integers. The only significant requirement that remains is that the sets Au ..., As be pairwise disjoint. Point processes in the spaces G, E, F, Af, etc. are also termed as line processes on U2, plane and line processes in (R3, segment processes on U2, etc. In general, we call random sets of geometrical objects which correspond to point processes geometrical processes. Remark A rich class of point processes arises if we allow the measure m in (7.5.2) to be random. A complete description of a process from this class requires the specification of the probability distribution of m. It can be useful to think about these processes as being constructed in two stages: first we draw a realization of m, and then we generate a Poisson process governed by this realization. For this reason such processes are called doubly stochastic Poisson (their other name is Cox processes). In stochastic geometry they have been considered in connection with Davidson's hypothesis (see [10], [43]). We consider processes from this class in §10.5. The construction of point processes by means of densities as outlined at the end of §7.4 for X = U generalizes to other spaces without substantial changes. We discuss densities for X = IRW case in the concluding sections of chapter 8.

7.6 The example of L. Shepp We give here an example due to Shepp of a point process in one dimension such that the number of points in any interval / is Poisson distributed with mean X\I\ for some k > 0, but for disjoint intervals these random numbers are not independent.

174

7 Stochastic point processes

We restrict ourselves to a point process *^( 0. Choose n with probability kne~n/n\9 n = 0, 1,..., and let Fn(x1,..., xn) = xx xn for n ^ 3 be the cumulative distribution function of the n points tl9 ..., tn of *^(
+ e(x1 - x2)2(x1 - x3)2(x2 - x3)2

x x1x2x3(l

-

Xl)(l

- x2)(l - x3).

(7.6.1)

For sufficiently small e > 0, F3 is a distribution function. Note that m{&) is not Poisson. Define Gn{a, b9 m) = Pn {exactly = C»PH{tl9...,

moftl9...,tne(a9b)} tm e (a, b) and tm+l9..., tn e (a, b)}

= C?Emf[(Ih(tj)-Ia(tj)) j=l

fl

(Ia(tj) + h(tj)-Ib(tj)\

(7.6.2)

j=m+l

where the probability Pn corresponds to Fn9 En is its expectation, and Ia(t)=l

if t < a

= 0 if t > a. Note that Wi)

lan{tn) =

Fn{au...,an).

In the expansion of (7.6.2) only terms of the form Fn(al9 ...,an) at = a, b or 1 for all i. Thus, if

Fn(au...,an)

= al

an

appear, where

(7.6.3)

for all such al9 ...9an then Gn(a, b, m) will be just as in the Poisson case. For n ^ 3, Fn is chosen to be uniform; for n = 3 (7.6.3) follows from (7.6.1).

7.7 Invariant models An alternative approach to the construction of probabilities P in the space {Ji, s/) is by using models. A point process is called a model in cases where we wish to emphasize that both the probability space (ft, !F9 p) and the map ^(co) are concrete (rather than 'general abstract' as in the definition in §7.1). Often, finding the values on #F of the distribution of a model can be too hard a problem. Then the map defining the model is the only method of descrip-

7.8 Random shift of a lattice

175

tion that remains. In bad cases even calculation of some expectations, onedimensional distributions, etc., can be a problem. The models we describe in the following sections all have distributions invariant with respect to a group. Definition: Let ^ be a group of transformations of the space X. A probability P on the corresponding (Jt, stf) is called invariant with respect to ^ (^-invariant) if for every Ae srf and every transformation (6 e ^ we have P(®A) = P(A\

(7.7.1)

where = {(Sm9 m E A}. A point process *e(co) is called ^-invariant iff its distribution is ^-invariant. By probability continuation it is possible to prove that P is ^-invariant whenever (7.7.1) holds for the events from the class Jf (see §7.5). Cases of invariance can be found among Poisson point processes. By virtue of (7.5.2), invariance of the governing measure m with respect to ^ implies invariance of the corresponding Poisson process with respect to the same group. For instance, if X = R", m is n-dimensional Lebesgue, then the corresponding Poisson point process is invariant with respect to the groups Tw, M n and A.. If X = G and m is the V f O 2 -invariant measure on G (see §3.6), then the corresponding Poisson line process in M 2 -invariant. Using the results of chapters 2-4 this list of invariant Poisson processes can be easily continued. In the three sections that follow we focus on invariant point processes whose nature is opposite to Poisson processes - those constructed by means of lattices of points.

7.8 Random shift of a lattice n

Let mQ c U be a lattice in R", i.e. a set of points in R" which in some 'affine' (non-orthogonal) system of coordinates have integer-valued coordinates (see fig. 7.8.1).

176

7 Stochastic point processes

We consider shifts t*n0 of this lattice. The shifts tl9 12 G Jn are termed equivalent if A set D c Tn is called a fundamental region (seefig.7.8.1) if (a) there are no pairs of equivalent shifts in D; (b) for every t eJn there is an equivalent shift in D. We choose for ft a fundamental region D, to be considered below asfixed,and let px be (the restriction of) the Lebesgue measure on D normalized by the condition PiO>) = 1We define a point process on R by means of the map n

»t(t) = tni09

teD.

(7.8.1)

In words: m,(f) is a lattice obtained from *^0 by a random parallel shift which is distributed uniformly within D. To see that m(t) is Tn-invariant we remark that to every event A c JfRn corresponds a set A* a D, namely A* = {teD:tmoe A}. Let P be the distribution of*n(t). Clearly where tA* denotes the usual shift of a set A* c D by a vector t, and tA*/D denotes the result of replacing each point of tA* which lies outside D by the equivalent point in D (seefig.7.8.2). Clearly, the transformation tA*/D is Lebesgue measure-preserving, therefore P(tA) = Vl(A*) = P(A). We leave it to the reader to prove that the distribution P of m(t) does not depend on the choice of the fundamental region D.

(a)

(b)

(c)

Figure 7.8.2 The point 0>2 is equivalent to ^ . The set tA*/D is the union of the four sectors in (c)

7.10 Lattices of random shape and position

177

7.9 Random motions of a lattice Let us first consider invariance with respect to rotations only. For ft we choose the group Wn of rotations of Un around 0. Let the probability p 2 on WB be proportional to the Haar measure on Wn. The map ^(w) = wm0,

(7.9.1)

where w*^0 is the result of rotation of *^0 by w e Wn, defines a Wn-invariant point process for any fixed m^sJi (not necessarily a lattice). Indeed, the probability of any A c JfRn equals

p2({weWn:wmoeA}). Our assertion follows from the relation { w e W r w*n0 G wx A} = w^1 {w G W n : w ^ 0 e A}. In fact a stronger proposition is valid. Let *^(co) be an arbitrary point process in Un defined on some probability space (ft, #", p). We construct a new point process m(w9

co) = w^((o)

(7.9.2)

which is defined on the product space Wn x ft where we consider the probability measure p 2 x p. The point process (7.9.2) is always Wn-invariant. Remarkably, this follows from a kind of Cavalieri principle, according to which p 2 x p is always invariant with respect to the transformations w x (w, co) -> (w t w, co),

wt G Wn

of the product space Wn x ft. Applying this remark to the process of §7.8 we conclude that wt<m0 (where *^0 is now a lattice, this process is defined on Wn x D) is Wn-invariant. The random lattice wtm0 is also ^-invariant. It is enough to show that for every t1 eTn the point process ty wt<m0 has the same distribution as wt<m0. This follows from the existence of t2 = t2(w) for which txw = wt 2 . We have and our assertion follows from the fact that the distribution oit2tm0 does not depend on t2 (coincides with that of t*nOi see §7.8). From the above two properties follows MM-invariance

7.10 Lattices of random shape and position In this section we consider only planar lattices whose fundamental region has unit area (1-lattices). A lattice m is called anchored ifOe*n.

178

7 Stochastic point processes

We can represent the space of all anchored 1-lattices by an appropriate subset D of the group A° (see §4.1); namely, each anchored m can be transformed into the standard square anchored 1-lattice by applying the following sequence of transformations of IR2 (the order is important). (1) A rotation w around O which brings the densest one-dimensional sublattice of m into a horizontal position. We can assume that O belongs to this sublattice. (2) A transformation H from the group H 1 (see §4.1) defined by the condition that horizontal one-dimensional sublattices in should be of unit space. (3) A transformation C from the group Cx (see §4.1) which is defined by the condition that be the square lattice. Under C the shift of the line y = 1 should be to the right (say) and of minimal possible length. This construction enables us to describe the space of anchored 1-lattices by means of the variables cp, h and c which correspond to the transformations w, H and C as described in §4.1. The domain D of these variables is as follows: 0 < q> < n f

4 0< c< 1

if h ^ 1

7(1 - /i4) < c < 1 - V(l - /z4)

if h ^ 1.

(7.10.1)

1/4

(Note that the value (4/3) can be obtained as the maximum of the minimal distance between the vertices of a parallelogram of area 1.) It is important to note that D a A^ is a fundamental region in A 5 in a sense similar to that of §7.8. Here two elements A°, A\ e A^ are equivalent if for some fixed m (the square 1-lattice say). To calculate the value of the Haar measure of D we represent dA° in the form (see §4.3) dA° = h~3 dc dh dq>. Then simple integration over the range (7.10.1) yields

L JD

dA° = \

(7.10.2) 6

i.e. the Haar measure of D is finite. Now we construct a model of a random A 2-invariant anchored 1-lattice. We take

7.11 Kallenberg-Mecke-Kingman line processes

179

p = p 3 , the restriction of the Haar measure dA° to D, normalized by the condition P3O>) = 1 and put m(A°) = A°m0,

A0 e D,

(7.10.3)

where ^ 0 is the usual square 1-lattice. The proof of invariance can be obtained by appropriate modifying of the reasoning presented by fig. 7.8.2. An A 2 -invariant model of (non-anchored) random 1-lattices can be constructed as follows (the group A 2 was introduced in §4.4). We take

a = (o, i) 2 x D, and we let the probability p 4 on il be the product of the normalized Lebesgue measure on (0,l) 2 and the p 3 considered above. Given an element (&, A0) e Q we define the shift t = 0 , A°0>9

i.e. t shifts O to the image of & under A0. Clearly t has uniform distribution in the fundamental parallelogram A°(0, I) 2 of A°m09 where ^ 0 is the standard square (anchored) 1-lattice. We now put m{», A0) = tA°m0,

(7.10.4)

0

where (^, A )GQ,. A proof of A2-invariance of this point process can be obtained by appropriate modification of the reasoning presented by fig. 7.8.2 since (t, A0) change in a 'fundamental region' of A 2 . Remark There is a connection between (7.10.2) and a theorem attributed to Siegel in [2]. The Siegel theorem states that Haar measures of fundamental domains in An are finite for every n > 0. The fundamental domains are defined with respect to lattices in Un. This implies the existence of An-invariant random lattices in Mn.

7.11 Kallenberg-Mecke-Kingman line processes This example was devised by Kallenberg, Mecke and Kingman [9], [48] to provide a counterexample to a hypothesis by Davidson (see [43]). The line process we describe is T2-invariant, does not contain parallel lines, cannot be represented as an m-Poisson line process with random m (is not Cox), and its second moment measure is locally-finite. By applying a random rotation around O, we obtain an M 2 -invariant line process which retains all remaining properties. Davidson's hypothesis was that processes with such properties do not exist. We use the representation of lines g e Gx (see §2.7) with coordinates

7 Stochastic point processes

180

tvg

AY Figure 7.11.1 g = (x, cot ifr).

(7.11.1)

Let us see how a parallel shift transforms the x, cot i// plane. We denote by tx and ty the projections of a shift vector t e T2 on the x and y axes. Clearly, tx and ty act on the line (7.11.1) as follows (see fig. 7.11.1): txg = (x + tx9 cot ^), while

tyg = (x-ty-cot

i/t, cot $).

Therefore the action of t on the x, cot \j/ plane reduces to, first, applying a transformation from the group Cx (see §4.1) with the cot \j/ axis, now playing the role of y axis, and, second, applying a shift parallel to the x axis. The order of these two transformations is not significant. The A 2 -invariant model of the previous section, if transplated onto the x, cot \j/ plane will be invariant with respect to the transformations just described (because the latter belong to subgroups of A 2 ). However, we are still not finished, since there are too many lines corresponding to our random lattice. That is, with probability 1 we have infinitely many points of the lattice projecting on any finite interval on the x axis. In the line reinterpretation, this means that with probability 1 infinitely many lines hit this interval. This contradicts the basic requirement that any m, e Ji^ should have a finite number of points in a bounded region of G. Thus what we obtain in this way is not a line process. The remedy lies in taking only those points from the random lattice which fall in some strip parallel to the x-axis. The (finite) width of the strip and the position of its central axis can be arbitrary (they become parameters of the line process). Thus the truncated random lattice will no longer remain A 2 -invariant. Yet it will retain in variance with respect to any subgroup of A 2 which maps the strip onto itself. As we have seen, the group T2 acting on the x, y-plane induces just such a subgroup acting on the x, cot i^-plane. Hence we have T2invariance of the line process whose lines correspond to the points of the random lattice which lie in the strip. We note that further examples of T2-invariant line processes can be obtained by applying appropriate independent thinning to the points of the A 2 -invariant lattice (the details are left to the reader). Sukiasian has constructed still further examples by taking parabolic inversion of the random lattice [47].

7.12 Marked point processes: independent marks

181

7.12 Marked point processes: independent marks To represent a point process in a space X we will sometimes write {xj instead of ^((o). This notation appeals directly to the concept of the countable set of points xf G X; the 'chance' variable co is suppressed. According to this a point process in a product space X xY can be represented as {(xt,yt)}>

(xi9yt)eXx

Y.

(7.12.1)

It is natural to call the random set { x j the projection on X of the above process. A projection can fail to be a point process. For instance, the set obtained by projecting on the x axis the points of a planar Poisson process governed by L 2 contains probability 1 infinitely many points in each interval on this axis. Definition: A point process {(xi9 yf)} in a product space X x Y is called a marked point process in X with marks in Y if its projection { x j is a point process. Let ^ be a group acting on X. A marked point process {(xi9 yt)} is called ^-invariant if the distribution of does not depend on © e &. In the group-invariant case it is possible under certain general conditions to define what is the distribution of typical mark; this idea plays a key role in the remaining part of the book and we give its definition in §8.1. Here we describe the situation in the simplest (but by no means most frequent) case of independent marks. We say that in {(xi9 >>;)} the marks yt are independent if (1) the point process { x j and the sequence {yt} are independent; (2) {yt} is a sequence of independent identically distributed random variables. The theory of the classes Jfx developed in §7.5 enables us to equivalently reformulate the above definition in terms of the point process in the product space X x Y which corresponds to {(xi9 yt)}. Namely we choose A e Jfx of the form ( I, Jx J r\

i-o and any sets Q Q c Y . W e define A* = An {the point from {xj which lies in It has its mark in Ci9 i = 1,..., s).

182

7 Stochastic point processes

The marks are independent if and only if

where Px is the distribution of some point process in X, while F is some probability distribution in the space Y. F is called the distribution of a mark and coincides with the distribution of the typical mark (whenever the latter exists, see §8.1). Now we give some examples of marked point processes with independent marks. I Poisson point process in a strip Let us denote by Sa the planar strip {(x, y): - oo < x < oo, 0 < y < a} = U x (0, a) c R2. Let ^(co) be a Poisson point process in U2 governed by the restriction of a Lebesgue measure X • L2 to Sa (equivalently, ^(
A* = An {the mark of the point from { x j which lies in / belongs to C}. Cleary A* coincides with the following subset of JfSa\ A*_(

B

i

B

2

#3

VI ' 0 ' 0 According to (7.5.2) we have

B2

Figure 7.12.1 The interval C is the projection of Bx on the y axis

7.12 Marked point processes: independent marks

183

where Px is the distribution of a Poisson process on the x axis governed by the measure Xa • L x . Because a similar factorization applies to every A e J^Sa, we conclude that in our example the projection process {xj is Poisson governed by Xa- Ll9 and the marks yt are independent with uniform distribution on (0, a). II Line processes The line process constructed in §7.11 could obviously be presented as a marked point process {(xi9 cot i/0} or {(x,., ^ ) } , where the points x( have been defined to lie on the Ox axis. In fact similar representations exist for any T2-invariant line process {gt} and any direction 6 if the condition P{there are lines in {gt} having direction 6} = 0 is satisfied. In this case we speak about ^-representation of {#J: by definition, in a ^-representation, { x j are the points of intersection of lines from {gt} with an axis having direction 0, \j/t is the angle between this axis and the corresponding line from {gt}. Let fi be a T2-invariant measure on G whose rose of directions m given by the factorization table 2.9.1 is atomless. What will be the ^-representations of the Poisson line process governed by ju? The answer is as follows For every direction 9, {x J is a one-dimensional Poisson process and {^.} is a sequence of independent angles; the measure which governs {xj is k{6) dx (Lebesgue) where X(0) is the rose of hits corresponding via (2.10.1) to the rose of directions of \i\ the distribution of each angle \\i{ is proportional to sin \j/ m0(dil/),

where me is the rotated rose of directions. To prove for instance Poisson property of {x J it is enough to take in (7.5.2) At = [/J = {g e G : g hits the interval / J , where / x , . . . , / s are disjoint intervals on the line which carries the {xj process, and to note that the sets At can be treated as pairwise non-intersecting. It follows that if a Poisson {gt} is governed by an M 2 - m v a r i a n t measure on G then the governing measure of {xj and the distribution of ^ cease to depend on 6; the distribution of \jf{ always has the density \ sin \jf dij/.

This corresponds to the property (7.15.7) below since such a line process happens to be Ml 2-invariant. Yet another way to obtain an M 2 -invariant line process is to take Poisson governed by a T2-invariant measure JX and then to subject it to an

184

7 Stochastic point processes

independent random rotation with uniform distribution on W 2 . The result will clearly be a doubly-stochastic Poisson point process (see the end of §7.5). It follows that on any fixed axis the process { x j in this case will be doublystochastic Poisson governed by random measure X(9) dx (where 9 is random and has uniform distribution on (0, n)). Also (generally speaking) the marks ^t in this process are not independent. A calculation of an expectation for this process is given in §10.5. Ill Poisson processes of balls A process {bt} of balls in Un can be described as a marked point process in Un9 namely {bi} = {(^,r £ )},

(7.12.2)

where ^ is the center of the ball bt and rt is its radius. We are interested in M n-invariant ball processes and here the simplest case is where {^} is Poisson governed by the Lebesgue measure X • Ln and the radii r{ are independent with some distribution p common for all centers. Such a process can be equivalently represented as a Poisson point process in the space Un x (0, oo) governed by the product measure X • Ln x p. Therefore in this case we call the ball process itself Poisson. Ball processes in (R2 are naturally termed as discs, and in U as segment processes. A natural condition usually imposed on such Poisson ball processes is that r w " 1 p(dr)< oo,

(7.12.3)

o which guarantees the following properties: (a) with probability 1 the balls of the process do not cover the whole of IRn; (b) on each line g0 a Un the 'trace' segment process is Poisson governed by some measure of the form X1L1

x px.

The expression of Ax and the distribution pl in terms of X may be performed in terms of p - a traditional topic in geometrical probability theory (see, e.g. [2] and [12]). IV Poisson cluster processes In a less trivial case X = Mn,

Y = Jt^n,

i.e. marks are realizations of point processes in Un. Let { x j be Poisson governed by X • L n , and let the marks yt = m,{ be independent. (This situation contrasts sharply with the one we face in our

7.13 Segment processes and random mosaics

185

approach to Palm distribution in chapter 8 where marks from JiUn will be strongly dependent.) We can consider the *^f's to be a sequence of independent identically distributed point processes in R": tni = *^((D).

If there exists a ball b(r, 0) such that P(*.,(©) <= b(r, 0)) = 1 and EPN{mi9 b(r, 0)) < oo then ^(co) is called a cluster. In this case the union «f(co) = (J r^f(co),

where tf = 0xi9 and *; belong to the Poisson process { x j , is a Tn-invariant point process; we call it a Poisson cluster process in IRn. If P(JVK(co), R") = 0 or 1) = 1 then *^(eo) will be Poisson. But if P(N(*^( 1) > 0 then ^(co) is non-Poisson. This assertion follows from the theory of §8.11. A similar construction can be used in the space G. We start from an fM32-invariant Poisson line process {gt} whose lines we now call 'parent lines'. To each parent line g( we attach a group (cluster) of lines parallel to g{\ let the probability law governing the number of lines in a cluster, as well as their distances from gi9 be the same for all parent lines; and let the realizations of clusters belonging to different parent lines be independent. We may assume that the lines of a cluster lie (with probability 1) within some strip of finite breadth. Then the union of all clusters will be an M 2 - i n v a r iant u n e process, which can also be called a Poisson cluster process. The theory of Palm distributions of line processes (see §10.1) can be an appropriate tool for their study.

7.13 Segment processes and random mosaics In most cases the T2-invariant line processes in U2 are completely determined by their marked point process representations. But this is not the case for a similar construction applied to segment processes. Let ^ ( c o ) be a Poisson segment process on IR2 (a point process on A2). It happens to be M^-invariant iff its governing measure \i is fVD ^-invariant. By Haar factorization fi(dS) = dM m(d/), where m is a measure in the space of segment lengths.

186

7 Stochastic point processes

To guarantee that the number of intersections of the segments from with any finite interval on the x axis be finite with probability 1, we require that

J

lm(dl) < oo.

(7.13.1)

Under this additional condition the marked point process {(xh i/^)} is well defined, where {xj is the point process of intersections of ^ 1 (co) with the x axis, and fa is the angle of intersection at xt. The process {(xi9 fa)} generated by ^(co) has a distribution of the same type as the process {(xi9 fa)} generated by the M2-invariant Poisson line process {gt} (see §7.12, II). For brevity we call this distribution PIAsin (Poisson, independent angles, sine law). Each line of Poisson {gt} is split into segments by other lines from {gt}9 and the collection of all such segments is a segment process ^ 2 (co). Clearly ^ 2 (co) is non-Poisson: with probability 1 ^ 2 ( » ) is a mosaic, while for ^(co) the probability of this event is zero (because the segments of ^(oo) display 'loose ends' which are excluded in mosaics, see §6.8, V). Thus the segment process *^2(to) is a random mosaic, and yet its {(xi9 fa)} process is of the same type as for Poisson ^(co), i.e. it is PIAsin. We construct now a random mosaic *^(
(1) ,

fflt

,

(2) 491 , . . .

be a sequence of independent random mosaics with common distribution identical to that of ^ 2 (co). Let {nk} be the collection of polygons generated by mi0\ For fixed k, the polygon nk is subdivided into an almost sure finite collection {nki} by the random mosaic Wfc). The random collection of polygons {nki} generates a random mosaic m(
Figure 7.13.1 Part of a realization of ^e(co). A is of X-type, B is of T-type

7.14 Moment measures

187

general the distribution of the process {(xh i/^)} by no means determines the distribution of the length of the typical (see §8.1) segment in an M 2 -invariant segment process. However, if a segment process is a random mosaic with no T-type nodes the situation changes. We give the corresponding formula in chapter 10.

7.14 Moment measures Conditions for the existence of different mathematical expectations underlie the treatment of many questions in stochastic geometry; we give some examples in the next section. Among these conditions the more frequent are the assumptions of existence of the first and the second moment measures. Let <m((o) be a point process in a space X, and let P be its distribution. As usual the random variable N(B9 m) equals the number of points of m((o) in a set B a X. We consider the expectation For every m e Jtx> N(B, m) as a function of B is a measure on X. Therefore mx(B) is also a measure on X but possibly not a locally-finite one. If mx happens to be locally-finite (i.e. a measure in our usual sense) then m1 is called the expectation measure or the first moment measure of m((o); the point process itself is of first order. Now let ^ be a group acting on X and let there be a unique (up to a constant factor) measure / i o n X which is invariant with respect of #. Clearly if<m((o) is ^-invariant and of first order then mx inherits the property of ^-in variance, and therefore necessarily m1=A'H,

0<>l
(7.14.1)

In such a case we say that (a ^-invariant) m(). The intensity has the meaning of the expected number of points from *«(co) in any B c X of unit /x-measure. The above ideas and terminology extend to random measures on X of a more general nature. Sometimes the random measures are generated by marked point processes in the same space X, and then one should be careful to distinguish between different interpretations of the finite intensity condition. We illustrate this by an example. Let m{&) be a T2-invariant segment process in U2. Suppose that using (2.13.3) we can describe m(vi) as a marked point process Mn) = {(Qi, '<)}• Here one of the possible finite intensity conditions is that the point process {Qi} be of finite intensity. The second is in terms of the random measure J?(m, B) = {the total length of the segments (or their parts) from <m which lie within B a U2},

7 Stochastic point processes

188

Figure 7.14.1 The circles represent the points of an m e Jfu\ the solid discs represent the points of <m2. Note the presence of points from ^ 2 on the diagonal {xx = x2}

and requires that EP<e = V L

with some

2

0 < Xx < oo.

(7.14.2)

Clearly the two assumptions say quite different things concerning *n(co) (neither follows from the other). Let us turn now to second moment measures. For every m eJfx we construct its 'square' m1 which is the following point set in X x X (seefig.7.14.1): m1 =

{(xhXj):xi9XjGfn}.

1

Clearly m e J?xx\', the dependence of m and co induces dependence of <m2 on co, thus we obtain a point process *^2((o) on X x X. We say that ^(co) is of second order if ^ 2 (co) as a point process on X x X is of first order. The expectation measure of *^2() be a Poisson point process in Un governed by k • Ln. If Bx n B2 = 0 then N(Bl9 *n) and N(B29 m) are independent. Therefore m2(B1 x B2) = X1' Ln(B1)Lm(B2)

if BtnB2

=0.

On the other hand, for every 5 c X , m2({(xl9

x2) :x1=x2e

B}) = EPN2(B,

m) = ALW(B) [ALM(B) + 1].

Thus by continuation m2 is a product of Lebesgue measures plus a singular component concentrated on the diagonal of X x X. A singular component

7.14 Moment measures

189

concentrated on the diagonal always appears in the second moment measure of a point process. If a second order ^ ( » ) is invariant with respect to a group acting on X then m2 inherits this invariance property. Therefore in appropriate cases factorizations in the style of our §2.14 or §3.15 can be applied to m2. Example 1 The unit radius disc process on IR2 which we consider in chapter 10 can be described as an M 2 -invariant second order point process (of disc centers) on U2. The assumption that the second moment measure m2 of such a process possesses a density means that outside the 'diagonal' m2 has a density with respect to L 2 x L 2 or equivalently, in the notation of (3.14.1), m2 has the form /(/)/ d/ dM ( / is called the density function). Example 2 Let {gt} be an M 2 -invariant second order line process on IR2 which has with probability 1 no pairs of parallel lines. Then m 2 = /i,

which is an M 2 -invariant symmetrical measure on G x G satisfying the conditions (6.7.9) and (6.7.10). In particular for the vertical windows vx and v2, as shown infig.6.5.1, we will have WiiiiQu 9i)' 0i ^ 02> 0i hi *s vl9 g2 hits v2}) = c I2,

(7.14.3)

where =

[(TC/2 - a:)

cot a + l]m(da) < oo.

(7.14.4)

The proof is by integration of (6.5.4); m is as in (6.7.9). Now we show that the convergence of the integral (7.14.4) is equivalent to the existence of the expectation of a random variable (which we shall encounter in §10.3). The argument we use is universal and therefore we outline it in general terms for a point process m((o) in a space X. For every set A a X x X we have c 1 ,x 2 )m 2 (dx 1 dx 2 )

Let f(xu x2) be a 'simple' function defined on X x X, i.e. where

/(*i,*2) = /s ASinAS2

=0

if(xux2)eAs9

if Sl # s2, U A, = X x X;

fs are any numbers. Multiplying the above equation for As by fs and summing, we get

190

7 Stochastic point processes

f(xl9 x2)m2(dx1 dx2) = EP

£

f(xh Xj).

(7.14.5)

By a standard argument of the theory of integration we conclude that (7.14.5) is valid for every measurable f We again consider the process of Example 2 and put r/7r_

Hr,o)9i^g2

^j

\

a C0

J

a

"I

J-

(The indicator equals 1 if the intersection point belongs to the disc b(r9 0) and a is the angle between gx and g2) Clearly (7.14.5) now becomes nr2 - c

= EP

IT? - <*
I

9ingjeb(r,O) \_\^

/

(7-14-6) J

where a0- is the angle between #£ and gj from a realization. Clearly c < oo (equivalently, {gfj is of second order) implies that the expectation in (7.14.6) is finite. Example 3 Let {sj be an M2-invariant second order segment process on U2. Its first moment measure is an fVO2-invariant measure of the space A2 = M2 x (0, oo) and necessarily has the form where m is a measure in the space of segment lengths / e (0, oo). The requirement that the process {^} of the sources of the segments be of first order is equivalent to the condition

3, oo)) m((0, oo)) = = f|

m(d/) < oo.

,00) J(O,G

The requirement that the process {xj of intersections of s/s with the x axis be of the first order is equivalent to the stronger convergence condition

f 00 / m(dl) < oo

Jo (the existence of the mean length of the typical segment in the terminology of §8.1). Let us look at the second moment measure m2 under the additional assumption that with probability 1 there are no pairs of parallel or antiparallel segments in {sj. Outside the 'diagonal', m2 is an IV02-invariant measure in the space A2 x A 2 \ {pairs of parallel or antiparallel segments} « M2 x S x x J1 x We use the notation

TJL

x (0, oo) x (0, oo).

7.14 Moment measures

191

§ ! for the space of angles \j/ between dx and <52, and ^ e A2; J1 for the space of one-dimensional shifts ti of St along the line gt which carries Shi = 1,2; (0, oo) for the space of lengths Jf of the segments. By Haar factorization m2(dd1 dd2) = dM m(d^ d/x d/2 dfx dt2)

(7.14.7)

with some measure m on the corresponding factor-space. If m2 has the form m2(dS1 dd2) = f2 dgx dg2 dtx dt2 d/x d/2

(7.14.8)

we say that m2 has density / 2 . M 2-invariance of m2 implies that f2 depends solely on the M2-invariant parameters: and we can reduce (7.14.8) to (7.14.7) by applying the transformation of §3.15,1. We find that m(d\j/ d/x d/2 dtt dt2) = f2 sin \\/ dij/ d/x d/2 dtx dr 2 . We say that { s j is of second order in hits if for every disc b(r, O) we have ^2({(^i, <52): both <$! and 62 hit b(r, O)}) < oo. In contrast with line processes, this is a separate condition which does not follow automatically from the existence of m2. If the process {s(} is Poisson governed by the measure dM /(/) d/

with

f

/(/) d/ < oo

Jo

Jo(7.14.8) equals then it is of second order in hits and f2 in f2=f(h)f{ll)Example 4 Let { s j be the process of edges of the mosaic on 1R2 generated by a Poisson line process governed by X dg ({sj = ^2(
=fcdMdpdl1dl2 = fD dM d/x d/2

on B onC on D.

192

7 Stochastic point processes B

Common endpoint

No intersection or common endpoint

0
p>0

Segments on a line Figure 7.14.2

/A> /B> fc a n d ID a r e functions depending on the corresponding M2-invariant parameters. The reader may easily find their exact form. Even without exact knowledge of the expression one can see that the value of m2({<$! hits vl982 hits v2}). (where v1 and v2 are the vertical windows shown in fig. 6.5.1) equals 4A2/2 (i.e. it coincides with (7.14.3) for the Poisson line process from which {sj was derived). Given a random mosaic, we say that the second moment measure m2 of its process of edges has a density if m2(d31 d<52) = f(5l9 S2)m°2(dS1 dS2)9

(7.14.9)

for some / which is called the density of m2.

7.15 Averaging in the space of realizations In §6.6 and §6.8 we derived a number of integral identities for functions depending on sets m ('realizations') from certain classes. The integrals were calculated with respect to Haar measure on groups. Suppose we have a geometrical process ^(co) which is invariant with respect to the group in question and whose realizations belong to the required class with probability 1, and in one of those identities we put

7.15 Averaging in the space of realizations

193

Averaging the result with respect to P, the distribution of ^(
i

N(*n, tB) dt = 0.

b(r,O)

We apply EP to both sides of this identity. By the Fubini theorem EP

N{*n9 tB) dt = Jb(r,O)

EPN(m9 tB) dt = 0. Jb(r,O)

By Tn-invariance EPN(*K, tB) = EPN(tn9 B) and therefore EPN(*n, B)LH(b(r, O)) = 0. We conclude that Ln(B) = 0 implies that EPN(*n,B) = 0. (7.15.1) We note that in the finite intensity case this result follows from (7.14.1). Because N(m9 B) is non-negative we also conclude that Ln(B) = 0

implies that

P(N(*n9 B) > 0) = 0.

(7.15.2)

II Let us additionally assume that ^(co) is of finite intensity X. According to (7.15.2), (6.6.7) and (6.6.8) are satisfied with probability 1. Therefore we can average these identities with respect to P. For the moment we suppose that the interchange of the order of the lim and EP operations is legitimate, i.e. that EP lim

It (m9 tD) dt = lim EP Jb(r,O)

It {m9 tD) dt

(7.15.3)

£ klk(m9 tD) dr.

(7.15.4)

Jb(r,O)

and EP lim

£ klk{m9 tD) dt = lim EP Jb(r,O) k = 2

Jb(r,O) k=2

Then by the Fubini theorem and the identity

194

7 Stochastic point processes

we will get that as D shrinks down to O

am

\

EPN{b(r,O),*n)

li

and 0

for/c>l.

(7.15.6)

To prove these important relations completely it remains to justify (7.15.3) and (7.15.4). For this purpose we note that for all t E b(r, 0\ and whenever the diameter of D does not exceed 1, we have N[*n9 tD) = N(m n b(r + 1, 0), tD). Therefore \\D\\

-1 I

N{m, tD) At = \\D\\~1 I

Jb(r,O)

N{m n b{r + 1, 0\ tD) At

JHr,O)

I 1

N(<mnb(r+ l,O),tD)dt

= N(m, b(r + 1, O))9 where at the last step we have applied (6.6.3). Clearly N(m, tD) = | klk(^ tD\ hence and £

klk(m9 tD) ^ N(4M9 tD).

k=2

Thus N(m9 b(r + 1, 0)) is an upper bound for both ratios

f

Ix(m9tD)dt

Jb( b(r,O)

and

i

X kIk{m,tD)dt.

b(r,O) k = 2

Since by thefiniteintensity assumption this bound is summable, Lebesgue's dominated convergence theorem guarantees the validity of (7.15.3) and (7.15.4). Thus the relations (7.15.5) and (7.15.6) are completely proved.

7.15 Averaging in the space of realizations

195

III The above examples make the essence of the method very clear: the Fubini theorem in conjunction with the observation that an appropriate in variance assumption concerning 4»(
where *^(co) is an fVD2-inva.ria.nt random segment process in [R2, and P is its distribution. For |<5| < 1 we have an inequality 1

[

1,(3, m)IfiW) d<5 < \S\-X [

Jb(r,O)

Ix{69 m) dd

Jb(r,O)

^ IS]'1 I

n(S, m) dd = IS]'1

Jb(r,O)

< \S\'X I

n(S, *nnb{r+

1, O)) d3

Jb(r,O)

n(S9 mnb(r+

1, 0)) d^ = 4 J ^ V ,

b(r + 1, 0)),

where if is the random 'length' measure introduced in §7.14; the last step was according to (6.8.1). We will have a summable upper bound, i.e. Ep&fa, b(r + 1, O)) < oo if the segment process is of finite intensity in the sense of (7.14.2). We come to the following result. For any M 2-invariant segment process *n((o) in IR2 of finite length intensity we have

~ f |si

= A- \S\ ~ f |sin tAi #

+ o(\S\).

(7.15.7)

Here 3 is any fixed segment in M2, f} is any Borel set of angles, and \j/ denotes the random angle between 3 and the segment from m((o) which hits 3. Here and below we use a slightly abused notation Downloaded from University Publishing Online. This is copyrighted material

196

7 Stochastic point processes

{8 is hit by k segments from ^(co)}

(6

. V

Lastly X = 2{nL2{B)Y1EP^{^

B).

We remark that if /? = (0, 2TT) then (7.15.7) reduces to (7.15.5). It follows that the length intensity Xx of m(
\S\~2 f

Ik(S,m)dS^

Jb(r,O)

X bi,bjEm, di,djhitb(r+l,O)

\S\~2

[ldt6j(5)d59 J

where hidjid) = 1 = 0

if ^ hits both 5t and 8j segments, otherwise.

If we additionally require that *^(eo) should not (with probability 1) possess pairs of parallel segments, then as follows from (6.8.4) and the above,

\sr2

Jb(r,C

,O) di,djhitb(r+l,O)

where O 0 is the angle between 5t and Sj. Thus the condition EP X (2 + (7T - 2* y ) cot « y ) < oo

(7.15.8)

bi,Sj 6 »?

^f^jhitbCr+l.O)

will suffice for our purpose. Note that (7.15.8) can be equivalently written in terms of the second moment measure of *^(
The result again refers to the intersection point process { x j , namely to

7.15 Averaging in the space of realizations

the probabilities of the events (

197

1 to have k intersections on fixed segment

\kj Let ^(co) be an M2-invariant segment process in U2 satisfying the condition (7.15.8). For every k ^ 2 exists the limit

lim \S\-2p(f) = y^EP |«5|->0

\K/

ZTt'Tir

£

Q(A/i).

(7.15.9)

Nieb(r,O)

Remark Condition (7.15.8) may be too strong. In fact it is not satisfied even in the cases of Poisson { s j or of { s j generated by Poisson lines as in §7.13 (the check is left to the reader), although in both cases the limits in question clearly exist. A proof of (7.15.9) can be given, however, under conditions which cover both cases. It is enough to require that { s j be of second order in hits and the existence of bounded density for the second moment measure of {s(} in the sense (7.14.8) or (7.14.9). The proof in the style of §§10.3 and 10.4 is left to the reader. In general (i.e. if no additional conditions leading to (7.15.9) are imposed) we can average (6.8.7) and use the Fatou lemma. This yields the inequality

lim inf \S\-2P( f ) ^ (in-nr^-'Ep |<5|-0

\kj

£

Ck(Nt).

Nieb(r,O)

Corollary If the intersection point process { x j on g0 is such that lim inf\d\-2P[ H-o

)=0

for all k > 3;

(7.15.10)

\k)

then the segment process possesses with probability 1 only nodes of order two (seefig.6.8.2). The proof follows from the fact that a point process on U2 of zero intensity is with probability 1 void of points and from the remarks in §6.8, II. VI In addition to the conditions of M]2-invariance, absence of parallel segments and (7.15.10), we assume now that ^(co) is a random mosaic. This clearly excludes the nodes of angle type (seefig.6.8.2) thus leaving the possibility of having order two nodes of only the remaining three types. The concrete random mosaics described in §7.13 fall into our class: in fact they possess nodes of cross or knot types but fail to display forks. We show now that under an additional general condition (7.15.11) the absence of forks is a consequence of the PIAsin property (see §7.13) of the marked intersection process {xi9 ^ } induced by a random mosaic on (any) line g0.

198

7 Stochastic point processes

The proof is done by averaging the identities (6.8.9) (which under our assumptions are satisfied with probability 1). It is not difficult to show that nN^r + 1, <m) (where Nx(r + 1, m) is the number of pairs of edges of the mosaic which hit b(r + 1, 0)) can be an upper bound for both ratios under the limit signs in (6.8.9) (assuming that |<5| < 1). Therefore the condition EPN1(r+ l,m)
(7.15.11)

guarantees that

lim \S\-2P(

M-o

£P[1 +(n-\n-fa-

fa\)

cot \n - fa -

W = (47i • Tcr2)"1 • (3EPnf + 4£ P n c + 2EPnk\

and \2 = (4TC • Tir2)"1 • (2nEPnf + 2nEPnc -h nEPnk). (In writing the left-hand sides we made use of the assumed independence of

{xj and {fa}.) The ratio of the two limits thus equals fl=

2EPnf + 2EPnc + EPnk 3EPnf H- 4£ P n c + 2EPnk

On the other hand, a depends solely on the distribution of the {^} sequence and therefore is the same for all random mosaics with the PIAsin property. But for a Poisson line mosaic governed by dg we have EPnf = 0,

EPnk = 0,

and (7.15.12) yields a = n/2. We can see from (7.15.12) that this can be possible only if EPnf = 0. VII More results concerning fVO2-invariant planar segment processes have been obtained by this method in [19], where the following problem was considered. Let m2 be the second moment measure of the random length measure J2? associated with a segment process ^(co) (see §7.14): m2(B1 x B2) = EPL(B1)L(B2)9 and let v2 be the second moment measure of the marked intersection process {(x;, fa)} induced by m(
7.15 Averaging in the space of realizations

199

distribution of the length of'typical' segment in m(
8 Palm distributions of point processes inUn

The idea of Palm distribution originated from the theory of point processes on the line [31]. For Rn9 connections of this idea with integral geometry have been identified in [32]. In this book our approach to the Palm distribution will be common for all spaces: we base it on Haar factorization. However, it is natural to present the simpler and more explored case of Rn in a separate chapter. The Palm distribution of a Tn-invariant finite intensity point process in Rn is often defined to be the conditional distribution of the process, given that the latter 'has a point at the origin O\ This can hardly be considered an honest definition since the conditioning event has zero probability. There are several equivalent rigorous definitions of this notion, but they can all be reduced to Lebesgue factorization. The importance of Palm distribution is rooted in the fact that, together with the value of intensity, it provides a complete probabilistic description of a point process which is alternative to that of chapter 7. In the concluding sections we concentrate on related analytical tools, especially 'Palm formulae' and consider certain equations that relate the distribution of a point process (in the sense of chapter 7) to its Palm distribution.

8.1 Typical mark distribution Let { ( ^ , kt)} be a Tn-invariant marked point process in Un (see §7.12) with marks kt belonging to some space IK. We say that { ( ^ , kt)} is of finite intensity if its {^} (which is a Tn-invariant point process in Un) is of finite intensity X. For a set A c Rn x IK we denote N(A) = card {i: ( ^ , kt) e A}.

8.1 Typical mark distribution

201

The first moment (or expectation) measure for {(0>h /ct)} is defined to be ml(A) = EN(A)

(8.1.1)

where E is the mathematical expectation. If { ( ^ ,fef)}is of finite intensity, then the values of mx on the product sets A = B x K,

5ci",

KcK

(8.1.2)

are finite whenever B is compact. Tn-invariance of { ( ^ , /cf)} implies Tw-invariance of the measure m1 (see §1.2). Hence by Lebesgue factorization mx = XLn x n ,

(8.1.3)

where II is a probability measure on K. U is called the distribution of a typical mark in the process { ( ^ , fct)}. Let us calculate mx on the sets in (8.1.2). We have

N(A)= X IM Applying (8.1.3) we get the following expression for II: E £

/*(*i)-

(8.1.4)

In particular, we see that the expression on the right-hand side does not depend on the choice of compact B a Un. We can obtain another useful interpretation for H(K) if we replace B in (8.1.4) by a sequence Bm of 'small' domains which converge to the origin O e Un. Then, as shown in §§7.15, II, we will have P (B™ ) = k • Ln{Bm) + o(Ln(BJ)

(8.1.5)

and

so that the limiting form (as m -> oo) of (8.1.4) will be

k is the mark of that point which lies in Bm. Let us look more closely at the ratio under the limit. Since, by the definition of k {k e K} < this ratio has the usual interpretation of conditional probability of the event Bm \ 1 has occurred. Thus (8.1.6) says that U(K)

(

can be defined as the limit of the corresponding conditional probability.

202

8 Palm distributions of point processes in U"

8.2 Reduction to calculation of intensities Given K c l K w e perform what can be called K-thinning of {^} (which is the projection of { ( ^ , kt)} on Un). Namely, a point ^ is deleted if its mark kt belongs to Kc (the complement of K) and is retained otherwise. The thinned set mK

= {&i:kieK}

(8.2.1)

is a point process which inherits the properties of Tn-invariance and of finite intensity. The number of points of the process (8.2.1) in a set B c Un can be represented as

I Hence it follows from (8.1.4) that the intensity of {^J* equals A(K) = m{K). Thus Tl(K) has a very simple expression in terms of intensities: U(K) = r ^ K ) .

(8.2.2)

This expression can be used for the statistical estimation of H(K) in cases where both X and A(K) can be reasonably replaced by numbers of points from and {^i}K within big volumes (i.e. in ergodic cases). Remark We did not assume above that the marks are independent. In fact our aim in the following sections will be to apply the definition of II to the cases where there is strong dependence between points and marks. In cases where marks are independent, however, and each is distributed according to some probability law F (as for instance in the examples in §7.12), both (8.1.6) and (8.2.2) show that

8.3 The space of anchored realizations Our approach to Palm distributions of point processes in Un as outlined in the next section is based on marked point processes with marks in the space which we call 'the space of anchored realizations'. By definition, Jl*^ = \m e JiRn: m has a point at 0 (i.e. is 'anchored')}. It can be proved [18] that Jf^n is a measurable subset of Jt^n (i.e. Jl%n e s$\ thus JC^n inherits a measurability structure from M^n. In particular, every measurable A c JtRn has a measurable counterpart A* in Jt^n, namely A* = A n J?£n. In fact any probability measure P on Ji%n can be considered as a probability on JiUn which has the property

8.3 The space of anchored realizations

203

P(Jt%n) = 1, and in this case of course P (there is a point of realization at 0) = 1.

(8.3.1)

Therefore on M%n no Tn-invariant probability measures exist (see §7.15,1). Each fixed m e Jt^n naturally generates countably many points in JC^n\ namely, if then, with each 3P{ e m, we associate the element fc. = ti*n9

where

tt = 9fl

(8.3.2)

(tt shifts m in such a way that ^ goes into 0). In other words, we map each m e JtUn into a countable system of points in the product space IR" x Jt^\ *.->{(^,fc,)}.

(8.3.3)

The number of points from the set {(0>i9 kt)} which fall inside a set A c Un x M%n depends both on A and the generating m,. We denote this number by

N*(A, m): AT*(X, m) = the number of points in A n { ( ^ , fe,-)}. (Note the essential difference between N* and the quantity N(B, m) defined in §7.1.) For certain sets A, the function N* can be represented as an indicator function of some sets in JKRn. The simplest are the following examples. Let D be a domain in Un. We put

and

A-\»

kY&eD ke(tC

where t = &0. Then it is an easy matter to check that N*(A, m) = IB(m). In fact the indicator of every set of the form

where each lt = 1 or 0, £ lt > 0, admits a similar representation i.e. IB(m) = N*(A, m)

for some A.

In particular this is true for every B e J^Rn (see §7.5) provided that £ /, > 0.

204

8 Palm distributions of point processes in U"

This observation has an important corollary in the theory of point processes, i.e. when we assume that in which case {(£?h k{)} becomes a marked point process in Un with marks in M%n.

It follows that probabilities of all sets from the class JfRn can in principle be determined from knowledge of the first moment measure EPN* of {(0>i9 &,)}. In view of the results of §7.5 this means in turn that EPN* determines the distribution P of <m((o) in a unique way.

8.4 Palm distribution The ideas of the previous sections can be successfully applied for the purpose of defining the Palm distribution of a Tn-invariant point process in Un of finite intensity L Let be such a process. We transform m((o) into a marked point process, *.(©)-> {(^,fc,)}, by applying (8.3.3) for each
(8.4.1)

From this and the closing remarks of the previous section we conclude that X and II together completely determine the distribution P of the point process m((o). In §8.8 we arrive at the same result from another point of view.

8.5 A continuity assumption

205

8.5 A continuity assumption Let us consider formula (8.1.6) in the context of Palm distribution, i.e. taking IK =

Jf^n.

We now choose the set K to be

(see §7.5), where D{ c Un are open domains whose closures do not contain the origin O. This K can be viewed both as a subset of JtRn and J?£n. The event {k e K} can be now written as

•••.* where t ^ is the mark of that point & e *n(
(

m

I lies in the domain Bm\ we put t = 0>O. Since £m converges to 0, the shift t is infinitesimal. Therefore we can expect that under some continuity condition we will have

A condition for the validity of the above can be derived from the conditions which guarantee (6.6.9). It follows from (8.1.4) that if

n

(o (

n

for any set Z <= U of Lebesgue measure zero which does not contain 0, then both conditions (a) and (b) in §6.6 are fulfilled with probability 1. We replace in (6.6.9) the ball b(e, O) by Bm and average in the space of realizations with upper bound N(m, b(r, O)) (compare with §7.15, II). Then using (8.1.5) we obtain

p((B;UB: lim m-oo

"

v

< Pi

m

\

*"

=(XLn(b(r,O))yiEP

£

hUtM

(8.5.2)

&>ie*nnb(r,O) V*/

By comparison with (8.1.4) we easily recognize on the right-hand side the value /D\ of Palm probability II of the event I - I. Thus (8.5.2) implies that this value \kj can be obtained as a limit of conditional probability (8.5.3)

206

8 Palm distributions of point processes in U"

The condition (8.5.1) can be given an equivalent form EnN(*n, Z) = 0

whenever Ln(Z) = 0,

(8.5.4)

in particular, the first moment measure of IT (if it exists) should be absolutely continuous, with respect to Lebesgue measure Ln. A sufficient condition for (8.5.4) can be given in terms of the second moment measure m2 of P (assuming m2 exists). According to (8.1.4) we have

= (XLn(B))-1EP

where m1 is the square of the realization m (see fig. 7.14.1) and 7* ^

( (Y \ /- [rpn Y — 11X j , -^2/ ^

Y

irp/i • Y c: R Y C: f *7\ * 1 ' 2 IZ-tX

where £ is the shift Ox. Hence (8.5.4) is equivalent to the condition that m2 on sets of the above type be zero. A sufficient condition for this (and therefore for (8.5.1)) is that 'outside diagonal' m2 should be absolutely continuous with respect to Ln x Ln (possess a density). Example If the second moment measure of a T^-invariant point process in R exists then it necessarily has the form m2(dxx dx 2 ) = dt m(dl) where dt is the Haar measure on Jl91 is the distance between the points x l 9 x 2 , m is a measure on (0, oo). Assume that Dl9..., Dsa R are intervals. Then Z reduces to a finite set of points and (8.5.1) will follow from the condition that the measure m(dl) does not charge any individual point on (0, oo). Recall that the set of such 'heavy' points can be at most countable. If (8.5.1) is violated, we cannot expect (8.5.3) to hold for every collection of domains D as described above. For instance, (8.5.2) does not hold identically in Example 1 of the next section. We note that (8.5.1) (and therefore (8.5.3)) hold for Tn-invariant Poisson processes in Rn. Condition (8.5.1) permits the modification of the right-hand side of (8.5.3), i.e. by replacing D by assuming that, as m -• oo, D(m) converges to D = (Du ..., Ds). Then a sufficient additional condition to have n(D

) = lim P

8.6 Some examples

207

can be that the interior of Bm contains O and does not intersect with the interior of any of the D(m)'s. However, 0 may lie on the boundary of one of the limiting domains Dt. The proof follows from an appropriate modification of (6.6.9). We use this possibility in the derivation of the Palm formulae in §§8.7 and

8.6 Some examples Example 1 Let m(&) be the point process described in §7.8. We have identically where tt has the form ^ O , ^ e t*n0. This means that marks (in the sense of §8.1) are non-random and coincide with ^ 0 . We conclude that IT is concentrated on the lattice *^0, i.e.

Example 2 Let m(ist) be the point process described in §7.9. For any tt = ^ e w£^ 0 , we have tiWtm0 = Wt-tmQ,

(8.6.1)

where t\ is the solution of the equation (see §3.5) Both sets in (8.6.1) contain O, therefore t[t<m0 = <m0. Thus for all i i.e. II is the distribution of the randomly rotated lattice m0. Example 3 Let P be the distribution of a Poisson process in Un (see §7.5). To find its FT we use (8.5.3). For any system of pairwise non-intersecting domains D = {Dl,..., Ds) such that 0 6 D , , i = l s,we get from (7.5.2)

This means that, outside 0, II is Poisson. Since always II (there is a realization point at O) = 1 the complete description of II in question will be n = A * P,

(8.6.2)

where A is the distribution of the point process consisting with probability 1 of a single point at 0, and * is the sign of composition (corresponding to the superposition of independent point processes). Example 4 Let ^(co) be the Poisson cluster process discussed in §7.12, IV. If the total number of points in each cluster has finite expectation, then ^(co) is

208

8 Palm distributions of point processes in U"

offiniteintensity. The Palm distribution II of *^(
8.7 Palm formulae in one dimension The derivation of Palm formulae in one dimension is directly based upon (8.5.3) and proceeds as follows. We consider the events happening in the intervals (seefig.8.7.1)

'-(4 and we assume the brief notations

6 2

5 2

o Figure 8.7.1

, 5 2

X +

8.7 Palm formulae in one dimension

209

Because the probability of having a point of realization at <5/2 is zero (this follows from (8.1.5)) we have 00

1= 0 k 1= 0

We are interested in summands which are 0(1) or 0(8) by order, as 8 - • 0 . Because of (8.1.5) the above can be rewritten as Pfc* - P o . f c

>X

puk

,X

O

^

^

In the present situation, (8.5.1) can be replaced by the condition II(there is a point of a realization at x) = 0

(8.7.2)

for every point x e U. Let us assume that (8.7.2) holds. Then using (8.1.5), (8.5.2) and the remark at the end of §8.5 we identify the limits which follow as values of Palm distribution: //n v \\ (8.7.3) (II is no longer Tx -invariant, and therefore when under II we show the interval completely). From (8.7.1) the limit of 8~1(pk(x + 8) — pk(x)) is thus found: dx where we have to put

Strictly speaking, we have shown above the existence of the one-sided derivatives d + p k (x)/dx. However, considering the intervals

-B and by making use of the remark at the end of §8.5 we can similarly derive the same expression for d~pk(x)/dx. Thus (8.7.2) guarantees the existence of dpk(x)/dx and the validity of (8.7.4) for every x. Equations (8.7.4) are called Palm formulae in differential form. They permit us to find the functions pk(x)

210

8 Palm distributions of point processes in W

in terms of IT:

CXf

'^"D-n^Jdu,

(8.7.5)

where the initial conditions are as follows: pk(0)= 1 iffc= 0 = 0

iffc>0.

Equations (8.7.5) are called Palm formulae in integral form. In contrast with (8.7.4) for their validity the assumption (8.7.2) is no longer needed. A proof of (8.7.5) for the general case can be based on the fact that (8.7.2) is satisfied for almost all values of x (if the second moment measure exists this easily follows from the remarks in the Example in §8.5). Thus (8.7.4) is valid for almost all values of x. It remains to show that the functions pk(x) are absolutely continuous. We consider their sums, Fk(x) = X Pi(x), i=k

which are monotone increasing in x (they may be interpreted as the distribution functions of certain 'waiting times'). We have

£ Fk(x) = £ kpk(x) = Xx. k=l

fc=l

Applying the Lebesgue decomposition theorem, we conclude that the functions Fk(x) do not possess discrete or singular components. Therefore the same is true for pk(x) = Fk(x) -

Fk+1(x).

8.8 Several intervals The approach of the previous section can be applied to the probabilities

where we assume that the intervals Il9 ..., Is do not overlap. The same arguments as used in the derivation of (8.7.4) yield a similar equation: dP

The derivation is taken with respect to the length of the rth interval; the shift tr sends the left end (say) of Ir to zero and I r = (0,..., 0, 1, 0,..., 0), where 1 occurs at the rth locus. The value of PI _ I now can be found by integration, \kj

8.9 Tj -invariant renewal processes

211

using the initial condition =0

if/c r >0

at/ r =O

/r 1 Ir+1

"

f

I

\\

if fc _

The event under P on the right-hand side depends on s — 1 intervals. Therehere I_ )

fore, by repeating the procedure we will end up with the expression of PI

\kj

in terms of IT and X. This confirms (in one dimension) the conclusion of §8.4. Remark The choice of the number r, as well as the endpoint of Ir in (8.8.1), was arbitrary. This leads (by means of integration of (8.8.1)) to several different representations for PI _ I. The independence of the result of integration of W the right-hand side of (8.8.1) is a necessary and sufficient condition for a probability n on J{0 to be a Palm distribution (see [50] and [51]).

8.9 J1 -invariant renewal processes There is an important class of point processes on a line IR which are called renewal processes. They are usually described as processes in which the intervals between consecutive points in a realization are independent identically distributed random variables. Yet, strictly speaking, this description is incomplete for it does not explain how, given a realization of interval lengths, we construct the corresponding set of points. In particular the situation is not clear if the process is intended to be Tx -invariant (which is of most interest for us). However, there is no difficulty in defining what is an anchored renewal point process. Let

...,£_2,£_1,£1,£2,...

(8.9.1)

be a sequence of independent positive random variables with a common distribution function F(x). Starting from 0, we plot the values in (8.9.1) as shown on fig. 8.9.1 and thus obtain a random realization <m e M\. For this point process probabilities of all events can in principle be expressed in

•f-i

o Figure 8.9.1 The solid points form a realization of the anchored renewal point process (there is a point of realization at 0)

212

8 Palm distributions of point processes in Un

terms of F. For instance,

where Fik) is the fcth convolution of F with itself (the distribution function of the sum £x + • • • + £k). In the case of finite intensity, a definition of a Jl -invariant renewal process can be as follows. Let *»(<©) be a point process on R, let it be Tx -invariant and of finite intensity. We call ^(co) renewal if its Palm distribution is the distribution of an anchored renewal process. Which anchored renewal processes can appear as Palm distributions for T^-invariant renewal processes? The answer to this question is that the corresponding F should possess a finite mean, i.e. A"1 =

x dF(x) < oo;

the condition is both necessary and sufficient (the reciprocal value X becomes the intensity of the J1 -invariant process in question). The following three-step stochastic construction yields the desired Jxinvariant renewal point process. Step (1) Sample a pair of random variables T_X and TX whose joint distribution function is given by the integral P{x1

>XUT1>X2)=1-X

!=

\-X\ \

(1 - F(u)) du.

(8.9.2)

Jo Step (2) Sample an independent realization of the sequence (8.9.1). Step (3) Plot the obtained values as shown on fig. 8.9.2. A proof of Tx-invariance of this construction would take too much space. We remark only that the special form of (8.9.2) follows from Palm formulae and therefore is necessary.

o Figure 8.9.2 The solid points form a realization of the T± -invariant renewal point process (there is no point at O)

8.9 Tj -invariant renewal processes

1

h~

213

1

1

Figure 8.9.3 With probability 1, 0 coincides with the left endpoint of a black interval

There is a class of segment processes on U which we also call renewal. Their loose description involves the following conditions: (a) the lengths of the segments of the process are independent identically distributed random variables; (b) the segments of the process do not overlap with probability 1; (c) the lengths of the gaps between consecutive segments of the process are independent identically distributed random variables. For convenience, we will call the segments of the process black and the gaps white segments. Let two distribution functions be given as Fb(x) = P{the length of a black segment is less than x} Fw(x) = P{the length of a white segment is less than x}. We sample two random sequences ...5-l9Sl9d29...:

independent lengths, distribution of each is Fb;

... v_l9 vl9 v 2 ,...: independent lengths, distribution of each is Fw. and plot them on a line starting from O as shown on fig. 8.9.3. We call the resulting process of black segments anchored renewal corresponding to the given Fb and Fw. Now an exact definition of a T^-invariant renewal segment process can be given. We use the idea of the relative Palm distribution (see §9.1 below for the corresponding definition). Given a process of (black) segments, by {/J we denote the point process of the left endpoints of the black segments. A Tx-invariant segment process for which {/J is of finite intensity is called renewal if its relative Palm distribution with respect to {/J coincides with the distribution of some anchored renewal (black) segment process. Example Let {sj be a Tx-invariant Poisson segment process on IR; it can alternatively be described (see §7.12) as a marked point process { s j = {(xh

\st\)}9

where the process { x j of segment left endpoints is Poisson governed by A • L l 5 and the corresponding segment lengths \st\ are independent and identically distributed according to some distribution function G(x). We assume that x dG(x) < oo.

(8.9.3)

214

8 Palm distributions of point processes in W

We construct the so-called Boolean model, which by definition is the random set

U={Js, If we think that each s( is colored black, then U is the resulting black subset of U; we call its complement, Uc, the white set. It can be shown that if (8.9.3) is violated then with probability 1 U = U; if (8.9.3) holds then U is a union of countably many non-contacting black segments which we denote as dt: U=\jSi9

dinSj = 0

ifi#j,

and in fact {£J is a segment process. The process {(5J of black segments is J^invariant renewal and we always have Fw(x) = 1 - e"**, where X is the intensity of {xt}. Instead of giving a full proof of the first assertion we demonstrate a somewhat lesser fact that the relative Palm distribution of {<5J with respect to {/J has the property that the parts of realization belonging to (0, oo) and ( — oo, 0) are independent. We split the space of segments (which is now IR x (0, oo)) into three nonoverlapping subsets I - the segments which lie entirely on the right of O, II - the segments which lie entirely on the left of 0, III - the segments which cover 0. The parts of {s J in these sets are independent Poisson processes governed by the restrictions of the measure X • dx • dG to the corresponding sets. The limiting (as e -• 0) conditional distribution of {s J , conditional upon the event {there are no segments in III} n {there is a left endpoint in (0, e)} has the form A,*^.

(8.9.4)

Here Ax is the distribution of a segment process which, with probability 1, consists of one segment whose left endpoint is at O with random length and is G-distributed; P1 is the distribution of the Poisson process on I u II governed by the restriction of X • dx • dG on this union. Because it is a Poisson process, its parts on the sets I and II are independent. The segments from the set II we represent now by means of their right endpoints y and lengths /. This corresponds to Cavalieri transformation y= x+Z /= /

8.9 T t -invariant renewal processes

215

which maps II into a reflected copy of the set I. The image of the measure will be again X dx AG. We conclude that Px corresponds to two independent Poisson segment processes, one on (0, oo) and the other on (— oo, 0) with distributions related via reflection. The relative Palm distribution of {<5J with respect to {/,} corresponds to the Boolean model of the segment process (8.9.4) and inherits the desired independence property from (8.9.4). We also deduce from the above remarks that the white length to the left of 0 has exponential distribution with parameter L Renewal Tx -invariant segment processes with exponential Fw also arise in connection with Poisson-Boolean models in many dimensions. Let, for instance, {bt} be an M^-in variant Poisson process of balls (see §7.12) in Un. The union of the balls from {bt}, i.e. the random set U = [j bh

(8.9.5)

is called the Boolean model for the ball process. It is again useful to call U the black set and its complement the white set. The trace of U on any line g0 is a J1 -invariant renewal process of black segments with exponential Fw. The proof follows from the observation that the process of chords bt n g0 is Jx -invariant Poisson. It is important to note that the above statements remain valid for Boolean models generated by rather general fMln-invariant Poisson processes of convex domains in Un. Are the mentioned properies of'trace' processes characteristic for PoissonBoolean models ? In chapter 10 we obtain some partial results concerning this question in the planar case. Our analysis there will involve two-dimensional marks which we attach to the black segments of the trace set induced on a line by the planar U as in (8.9.5). Namely, we consider the marked segment processes where the (black) segments St are the non-contacting components of the trace set U n g0, and the mark (i/^-, i//") consists of the angles shown on fig. 8.9.4.

Figure 8.9.4 The angles are between g0 and dU at the endpoints of the black segments

8 Palm distributions of point processes in Rn

216

In this situation the above can be complemented by the following statement. If a black set U is the Boolean model for an M 2 -invariant Poisson disc process then the triads (\dt\9 ^-, iAD f° r different values of i are independent. (More exact formulation should, of course, be in terms of the anchored version of the process {(Sh \\i[, *A0})- Independence of the pairs (^, \jj") with different f s follows from the independence of the elevations of the centers of the discs from {bt} which actually hit g0 above this line.

8.10 Palm formulae for balls in M" Let bv be the ball of volume V centered at O. We consider the annuli a

o r

v,h = bv+h\by

a

v,h =

bv\bv-h-

As mentioned in §6.6,fig.6.6.2, (6.6.7) and (6.6.8) remain valid if we replace D by these annuli and let h tend to zero. By averaging in the space of realizations we conclude that (7.15.5) and (7.15.6) remain valid if D is replaced by our thin annuli. By an easy modification of the method which led to (8.5.2) we can show the existence of the limits (see [32])

The functions n above may be interpreted as the conditional probabilities of the event II

y

I, the condition being that there is a point of ^(co) on the

boundary dbv:

f n(

tbv

)dt*.

(8.10.1)

Here tbv is the shift of bv by the vector t e dbv (so that always O e d{tbv)\ dt* is the area measure on the surface of bv, and \dbv\ is the total 'area' of dbv. In other words, t is distributed uniformly over dbv. By considering events which happen in bv+h and bv and applying essentially the same argument as in derivation of (8.7.4) we come to the desired Palm formulae dps(V) dV

lr

- ns(V)l

(8.10.2)

where

The derivation of formulae similar to (8.10.2) for several balls or n-dimensional intervals is left to the reader. We use (8.10.2) in the next two sections.

8.11 The equation II = 0 * P

217

8.11 The equation II = 0 * P In this section we consider the relation n = 0*P

(8.11.1) n

as an equation where 0 is a distribution of some point process in U and is assumed known. The problem is to find a distribution of a Tn-invariant point process in Un of finite intensity whose Palm distribution II can be calculated by means of (8.11.1) (the composition sign * corresponds to the superposition of independent point processes). We saw in §8.6 that Poisson cluster processes satisfy (8.11.1); hence, at least for some distributions 0 , (8.11.1) indeed has solutions. Let us show how the probabilities p(

1 can be found for all balls

from (8.11.1). We introduce the following generating functions:

TT(Z, V) = X nk(V)zk; (the probabilities nk(V) have been defined in (8.10.1)). Equation (8.11.1) implies that II(Z, tby) = 0(Z, tby)P(Z9 by) for every shift t. Integrating this over dbv and using (8.10.1) yields TT(Z, V) = P(z, by)G(z, bv\ where 0 ( z ,fcK)= \dby\~1

0(z, tbK) dt*.

Let us formally apply (8.10.2), which in terms of generating functions is now

written as ^

^

, by).

Its solution satisfying the initial condition P(z, bv)\v=o = 1 is clearly

Cv ~ P(z, by) = exp{A(z - 1)

0(z, bu) dw}.

(8.11.2)

Jo Of course this is a partial result since for a complete description of P we need

218

8 Palm distributions of point processes in U"

to find the probabilities of the type % where the /f's are n-dimensional intervals. We see from (8.11.2) that the solutions of (8.11.1) cannot be unique (if they exist); in fact we have to deal with families of solutions depending on the parameter k (the intensity). Example Let us consider the equation II = A * P ,

(8.11.3)

where A is the distribution of a point process which has only one point in the whole of IRW placed at O. In this case ®(z, tbu) = 1

for t e dbu

®(z, bu) = 1, and from (8.11.2) we find that P(z, bv) = exp{k(z - l)V}.

(8.11.4)

By applying n-dimensional versions of (8.8.1) we can show that (8.11.3) has only Poisson solutions.

8.12 Asymptotic Poisson distribution In this section we consider a rather special but important feature of a point process - the distribution of the number N(m, bv) of points within a ball bv. In fact to derive the Poisson property of this variable (i.e. (8.11.4)) much less than (8.11.3) is needed as (8.11.4) follows merely from the equation PsiV) = ns(V).

(8.12.1)

We repeat that {ns(V)} may be interpreted as the conditional distribution of the number of points in bv, given that there is a point of the process on dbv, see (8.10.1). Can we make any conclusions concerning the distributions in (8.12.1) under less stringent assumptions then their coincidence? We show now that the Palm formulae (8.10.2) make this possible. We rewrite (8.10.2) in the form dPs(V) = *tP,-i(V) - PsiV)-] + kld^iV) dV where SS(V) = ns(V) In terms of the generating functions

ps(V).

- Ss(V)l

(8.12.2)

8.12 Asymptotic Poisson distribution

219

(8.12.2) becomes dP V)

f

z, V) + 2(z - l)8(z, V)

(8.12.3)

or, after solving with respect to p(z, V), I

A

Jo

- " ^ ( z , II) dw.

(8.12.4)

The characteristic function of the random variable Njm, by) - XV equals f(limp(eunjim9

V)

and therefore can be obtained from the right-hand side of (8.12.4) by substituting and multiplying the result by ei

-\\

[V \5(z,u)\du^l\z-l\

Jo

=x\z-\\

Jo

r^^sM-Ps

rP(u)du. Jo

The function is the variational distance between the distributions {ns(u)} and {ps(u)}. Since as V -• oo we have I /"A

ZWl + the required condition will be

r

p(u) du = oQV) as K -^ oo. (8.12.6) Jo Jo is a sufficient condition for rjv as defined by (8.12.5) We have found that (8.12.6) to be asymptotically normal distributed with zero mean and unit variance. Due to the specific form of (8.12.5) (normalization by the square root of

220

8 Palm distributions of point processes in U"

expectation) this property can be called the asymptotic Poisson property of the distribution of N(m, bv). In fact (8.12.6) imposes a restriction on the rate of the decrease of the distance p at infinity (it must be not too slow). Loosely speaking, if the condition of having a point on the boundary of bv does not affect the distribution of the number of points in bv for larger values of V too much, then N(<m, bv) is asymptotically Poisson distributed [52].

8.13 Equations with Palm distribution The success of (8.11.3) suggests that we should try to apply similar equations in more complicated stochastic situations. Below we introduce two different equations of this style. In the next section where we describe their solutions the interrelation between the two problems will become apparent. I A natural idea is to replace P in (8.11.3) by the result P* of some operation acting on P. Here we will define P* as obtained from P by means of location dependent but statistically independent thinning. More properly let us assume that a function h(0>\ 0 ^ h(gP) ^ 1 is defined in Un. Given a realization <m we define ^ * to be a random subset of m\ a point &i e m belongs to m* with probability /i(^); for different values of i the events {&i e *n*} are assumed independent. Let P be the distribution of a point process *^(co). Then by P* we denote the distribution of the thinned point process ^*(«). The desired version of the equation (8.11.3) will be II = A*P*.

(8.13.1)

Here the problem is to describe conditions on k and on the function h(^) which ensure the existence of a Tn-in variant probability P of finite intensity for which (8.13.1) holds as well as to clarify the uniqueness questions. II The second equation provides a description of processes of non-intersecting balls of unit radius which otherwise 'do not interact'. These two seemingly contradictory requirements - non-intersection and non-interaction - can be reconciled as follows. We assume that the centers of the balls constitute a Jn-invariant process m(
8.14 Solution by means of density functions

221

Ax = {there is a point of *^(
and B(2) is the ball of radius 2 centered at O. We will write P(-/A) for the distribution of<m(
(8.13.2)

We postulate that the mathematical expression of the 'no interaction' property is that 'outside 0 ' we have P(-/A1nA) = A*P(-/A).

(8.13.3)

Intuitively speaking, this equation means that if there is enough space somewhere in the realization to put a ball there (condition A), the actual placing of a new ball (condition A n A^) does not affect other balls of the realization. Equations (8.13.2) and (8.13.3) can be put together to give n = A*P(-/A),

(8.13.4)

which is another direct generalization of (8.11.3).

8.14 Solution by means of density functions I First we consider the equation (8.13.1). We are looking for the solution P in the class of point processes in Um generated by continuous densities (p(xl9..., xn\ see §7.4. We will use the interpretation (7.4.7) with dxt now denoting infinitesimal volumes in IRm. Note that because of Tm-invariance and by (7.15.5) we have cpiXi) = A,

the intensity of P.

How are the density functions of the distributions II and P* expressed in terms of the functions cp(x1,..., x j ? It follows from the construction of P* by means of thinning that

and we conclude that the density functions for P* are necessarily of the form n

q>*(xl9 . . . , * „ ) =
222

8 Palm distributions of point processes in W

Denote by (pn(x1, ...,xn) relation dx,

the density functions of II. We base on a heuristic dxn\

„ m-ln(d0

dxt

dxr

which is a version of (8.5.2) in which the domains D( are taken infinitesimal and which can be easily justified under the assumption (7.4.9) concerning densities cp. This implies cpn(x1,...,

xn) = A~V(0, xx,...,

xB).

(8.14.1)

By Tm-invariance of P
= q>(0,x2 -xu...,xn+1

-xj

= k(pu(x2-xu...,xn+1

-xx).

The equation (8.13.1) implies the coincidence of densities cp* and q^ i.e. n+l

We easily derive from this that
(8.14.2)

In order to guarantee that q>(xl9..., xn) be symmetrical functions it is enough to require (and we do) that h(-x)

= h(x).

Also the condition is satisfied because 0 ^ h ^ 1. Thus it remains to find out when condition (2) of §7.4 is satisfied. For then the corresponding P will become a solution of (8.13.1) and unique within the class we consider. The following proposition whose complete proof can be found in [64] provides the answer. Assume

r = 1* (1 - h(u) u)) du < oo. J lrcm

Then for the values of k in the interval 0 < k < (eC)"1 (e is the base of natural logarithms) (8.14.1) are density functions of a Tm-invariant point process of finite intensity L Its distribution P satisfies (8.13.1). In one dimension the case fO if|x|
8.14 Solution by means of density functions

/(0,x)\ = fl

ifx
i, o ;

ifx>i.

Wx-D

223

Substituting this in (8.7.4) for k = 0 we get dpo(x) dx

= -kpo(x - 1)

if x > 1

(8.14.3) po(x) = 1 - kx for 0 < x < 1. The above statement implies that the solution of (8.14.3) remains non-negative whenever k < e"1. This particular result was obtained in [65] with no probabilistic interpretation. II Instead of (8.13.4) we prefer to consider a somewhat more general equation IP(d^) = c 6 W ? ( d « ) (8.14.4) which expresses the idea that II has a density c®(m) with respect to P. Here again P is a Tm-invariant probability on J R m of finite intensity k, II* is obtained from the Palm distribution of P by discarding the point at O. We obtain (8.13.4) from (8.14.4) when we take &(m) = 1

if m e I ^ I, and 0 otherwise

Vo / and

For a given function 0 ( ^ ) we can expect a family of solutions for (8.14.4) (if any exist at all) and the constant c depends on the particular solution we choose since it has to satisfy the norming condition

-I-

(m)P(dm).

(8.14.5)

We try to find the solutions of (8.14.4) in the class of probabilities P generated by continuous densities
l

0W?(d4

(8.14.6)

I the variable m can be represented as

and on the same set P(dm) can be written as which is a heuristic analog of (8.5.3). Here II* is the image of II* under the shift map

224

8 Palm distributions of point processes in W

<m - • tx*n

where

tx = Ox

and satisfies the equation n * ( d ^ ) = ®x(™)P(dm)

(8.14.7)

where Using (8.14.1) we find further

2 fH

J °^ U X

where of course 0 O = 0 . Replacing ring here her 0 by xx and xx by x 2 we obtain (/)(xl9 x 2 ) = (ck)2

®Xl(™ u x 2 ) 0 X 2 W P ( d 4

Acting similarly we find a general expression

•Xl(*« u {x 2 ,..., x n })0 X 2 (^ u { x 3 , . . . , : (8.14.8) Because of (8.14.5) this yields in the case n = 1 (p(xx) = k. The relation (8.14.8) remains unexplored for general 0 . We will concentrate on the case (which includes (8.13.4))

®x(m) = PI h(x - ^ where h is an even function defined in Um. Clearly in this case for; = 1,..., n

®x(*nv{Xj+l9...9xn})=

f[ h(xj-xk)' k=hl

n

hixj-^i).

Pi*™

Therefore (8.14.8) reduces to
(8.14.9)

where a(xl9...,

xn) = k\ [ ] h(xj - xk),

kx = ck,

n

h (x

x \ — rr rr h(x — w\

°m\Xl-> ' ' ' 9 Xn) — 1 1 1 1

n X

\ j

*^ih

If kx and h satisfy the conditions of the proposition in part I of this section

8.14 Solution by means of density functions

225

then a(xl9 ..., x j happen to be density functions of a probability P o on J R m . Because now 0 ^ h(x) ^ 1 the products a(xl9...9xn)bm(xl9...9xn)

receive interpretation of density functions of a point process which is obtained from P o by independent thinning. The thinning procedure is as follows. Let mQ(is>) be a point process whose distribution is P o . A point Qx e **o(<°) survives with probability Yi^e^HQi — &j) (this probability depends on m) and for different points Qh Qs e *n0 the events of survival are independent. The distribution of the thinned process (i.e. of the points which survive) we denote by P^\ Then (8.14.9) can be replaced by an equivalent relation P= \P(r>P(dm).

(8.14.10)

This equation has the advantage that it can be solved by successive approximations. Namely we start with some Px (it can be Tm-invariant Poisson), and define

pn+1 = [p^pn(dm),

«=1,2,....

The problem is to show that the sequences of probabilities Pn converges to a limit P which is a solution of both (8.14.10) and (8.14.4) and which does not depend on the choice of P x . Conditions for such a convergence were found in [67]. We now complement the above by a brief outline of an explicit result concerning (8.13.4) in one dimension. We consider the functions

We assume that no(x) is continuous. Then the Palm formulae yield

J ^ M = -Ano(x).

(8.14.11)

dx The basic relation (8.13.4) implies 1

ifx<2 Po(4)

ifx>2.

Therefore for x > 2

7io(x + 2) = - ^ p ^ ^ o W -

(8.14.12)

This relation enables us to determine no(x) for all values x > 4 if 7co(x) is in

226

8 Palm distributions of point processes in U"

some way given on the interval (2, 4). But no(x) cannot be arbitrary on (2, 4). For every x e (2, oo) and every k > 0

The function 7io(x) must remain non-negative. It follows that no(x) should be absolutely monotone on (2, oo); in other words a measure \x on (0, oo) exists such that

,-J.

The values on a half-axis of Laplace transformation of a measure defines the latter uniquely. Therefore we conclude from (8.14.12) that for every interval / c (0, oo)

f ae2«/i(da) = - ^ r f Mda). Ji

(8.14.13)

Po(V Ji

The equation e2a = const cue has only one solution. Considering (8.14.13) for small intervals / we conclude, that fi is concentrated on a single a which is the solution of the equation X This means that

^ W = {ex P ; f -a(x-2)) if« > 2.

(8 R14)

-

In fact we can choose a > 0 arbitrarily. Then the parameters X and po(4) should be determined from the equations l

=

JJ0

no(u) du = 2 + a " 1

i=l-/l

(8.14.15) TTO(W) dw

Jo A deeper analysis (see [66]) shows that the point process in question happens to be a renewal point process with the distribution of the space between the consecutive points given by (8.14.14). It corresponds to a segment process which is renewal with constant segment length 2 and exponential distribution of intersegment spaces (see §8.9).

Poisson-generated geometrical processes

One application of Haar factorization leads to the concept of relative Palm distribution of a group-invariant geometrical process. By associating different point processes on corresponding groups, we obtain different Palm-type distributions for the same geometrical process. When the geometrical process is Poisson, its relative Palm distributions can be calculated, and the result is usually reminiscent of (8.11.1). Much of this chapter will be devoted to different corollaries of such results. The purpose is to derive size-shape distributions (or at least to give the corresponding stochastic constructions) of typical configurations generated by Poisson processes.

9.1 Relative Palm distribution Let X be a group of transformations of a space Y. For instance, X can be Tn, M n or AB, n = 2, 3 ... and Y can be the corresponding Un, or G, r , E etc. We write xeX.yeY. Suppose we have two jointly distributed point processes **!(©) c X

and

^ 2 (co)czY.

(9.1.1)

We can speak of simultaneous transformations oiboth any x e X : We assume below that the joint distribution of the processes (9.1.1) is invariant, that is for any x e X the pairs ^l(C0), m2((O)

and

Xtn^Q)), X<m2((O).

possess the same joint distribution. Another basic assumption is that <m1 (a>) is of finite intensity. This means that the mean number of points from ^ ( o ) in a domain D c X i s XH(D\ where H is the Haar measure on X (in all cases we consider, H is actually bi-

228

9 Poisson-generated geometrical processes

invariant). If both assumptions are fulfilled, we can construct the relative Palm distribution of ^2(co) with respect to ^(co). We base this construction upon Haar factorization. We consider a marked point process on X: {(xi9 xT^yt})} where {xj = ^ M , {)>,} = ~2(<») (9.1.2) with marks belonging to the space of realizations My. Let us consider the counting function N(Bl9 B2, co) = the number of points x( e ^1(co) which lie in B1 a X and whose marks lie within B2 <= My. The marked point process (9.1.2) is invariant with respect to X. This property is inherited by the expectation of N, therefore we have the Haar factorization (see §1.3) EN(B19 B29 co) = WiBJU^fo),

(9.1.3)

We choose the value of X in (9.1.3) from the condition that H^i be a probability distribution, in which case X coincides with the intensity of {xj. We call the probability distribution Hmi on My defined by (9.1.3) the relative Palm distribution of {y(} with respect to {xj. We rewrite (9.1.3) as

•JJ'

=E

where we denote by m2 the elements of My. Using linearity, we derive from this the following: V(x, m2)H(dx)nmi(dm2)

= E X V(xh x r ^ 2 ) ,

(9.1.4)

Xtemi

first for the functions V which are linear combinations of indicator functions or IBlXB2 type. Then by standard measure-theoretic argument we establish that (9.1.4) is valid for any measurable function V which maps X x My onto the real axis (compare with (7.14.5)). We get the simplest example of a Palm distribution as defined by (9.1.3) when we assume that Y = X and that ^ 2 ( ( 0 ) = ^i(co). In this case Hmi = IT defines what can be called the Palm distribution of a left invariant point process {xj on a group X (assuming that {xj is of finite intensity). The general topic of point processes on groups (especially non-Abelian ones) remains largely unexplored. In the sections that follow, processes on groups appear merely as specific tools for the study of particular geometrical processes. We start each time with a geometrical process *n2(
where

m2 e My, <mx e

M\-

Whenever <m2 happens to depend on co (is random) a mapping of this kind

9.2 Extracting point processes on groups

229

automatically induces dependence of *n± on co. Hence the joint distribution of the processes m2(
9.2 Extracting point processes on groups The following examples will make clear the general idea of how a point process on the group IVO 2 can be associated with a given geometrical process in R2. A case where we 'extract' a point process on T2 is considered in §§ 9.3 and 9.4. Example 1 Let {gt} be a line process on IR2 (a point process in G). On each line gt of the process we plant an independent Poisson point process each governed by Lx (the one-dimensional Lebesgue measure). Each of the points thus obtained together with the direction of the carrying line g{ determines a motion MseM2 (see §2.13). These motions constitute a point process on M2 which we represent as {Ms}. The pair of dependent processes will be **!(©) = {MJ,

**2(«) = {gt}.

The property of M 2-invariance of {gt} directly implies invariance of the pair m^Gjl) and m2((o) in the sense of the previous section, and {Ms} happens to be of finite intensity whenever {gt} is of finite intensity (see §7.14). In the case {gt} is a process on G (nondirected lines) we can apply the same construction by converting gt into a process on G. This can be done by random equiprobable assignment of one of the two possible directions to each gt; the choice of the directions for different lines can be independent. Example 2 Again let the geometrical process be Let us assume that with probability 1 there are no pairs of parallel lines in {gt}. Then each pair of lines gt and gj has a single intersection point Qij = 9i^

Gy

For each Qtj we construct four unit length directed segments as shown ia fig. 9.2.1. We denote these segments by M s . Again each Ms can be identified with an element of the group IU2. The obtained random set of motions {Ms}'

230

9 Poisson-generated geometrical processes

Figure 9.2.2 The segment Ms emerges from a point on a dbt in the direction tangent to dbt. The disc bt remains in the left halfplane with respect to continuation of Ms

is the desired process on M 2 ; The conditions of §9.1 will be satisfied whenever {gt} is fVD^-invariant and of second order. In the above examples the marks M[1m2 are different in nature. In both cases they contain the x-axis. In the second example, M^<m2 necessarily contains yet another line through 0 whose direction is not specified and IT^i can be considered as a distribution in the space where the circle S x is appropriated for the direction of this line. Example 3 Let {bj be a disc process in (R2. On the boundary of each bt we plant an independent Poisson process governed by the arc length measure. Each of these points determines a directed segment (equivalently, a motion) Ms e M 2 in the manner shown in fig. 9.2.2. In this way we obtain a point process {Ms}" on fVO2 a n d w e P u t **!(©) = {Ms}f\

*n2(
We have the situation described in §9.1 if {foj is M 2-invariant, the process of centers offers is of finite intensity and the typical radius distribution (see §8.1) has a finite mean. Each M~1<m2 contains a disc tangential to the x axis at 0 which lies in the upper halfplane. Example 4 Let {sj be a segment process on (R2. With each st from a realization we associate an M{ G M 2 (we again think of Mt as a directed segment): the source of M( lies at the midpoint of st; the direction of M( coincides (mod n) with the direction of st. We choose at random, with probability 1/2, each of the remaining two possibilities to fix the arrow on Mi9 according to the outcome of an independent experiment of tossing a coin. We put If {sj is M 2-invariant and the process of the midpoints is of finite intensity then the conditions of §9.1 will be satisfied. Each M^lm,2 will contain a

9.3 Equally weighted typical polygon in a Poisson line mosaic

231

segment on the x axis with the midpoint at O. We note that n ^ t can be considered as a distribution in the space (0, oo) x

Jt^,

where the factor space (0, oo) is appropriated for the length of the mentioned segment. We leave it to the reader to prove that the projection of H^i on (0, oo) coincides with the distribution of the length of the typical segment in {sj (see §8.1).

9.3 Equally weighted typical polygon in a Poisson line mosaic Let {gi} be a Poisson line process on U2 governed by an PVD^-invariant measure on G (i.e. by X dg). The lines of {g{} split the plane in non-overlapping convex polygons nil thus the collection {TCJ is a planar random mosaic (see fig. 9.3.1). Problem (Loose formulation): find the 'distribution' of the polygon randomly chosen from the collection {TTJ. The choice should give 'equal weight' to 'each' polygon of the mosaic. We called the above formulation 'loose' because in the present situation the notion of random choice 'with equal weights' is ambiguous: the number of polygons nt is infinite with probability 1. Also, because of the absence of uniform distribution in the whole of R2, we can at most hope to obtain a probability distribution of the Tn-invariant characteristics of the randomly chosen polygon. Therefore it is natural to look for a solution in the space of 'anchored' polygons. Below we call a polygon n anchored if (1) the origin O e U2 is a vertex of n; (2) n has no other vertices on the y = 0 axis; (3) the interior of n lies on the right of the y — 0 axis (see fig. 9.3.2). We put K = the space of anchored polygons.

Figure 9.3.1

(9.3.1)

232

9 Poisson-generated geometrical processes

Figure 9.3.2

Using the ideas of §8.1 it is possible to reformulate the above problem so as to give it an absolutely precise meaning. We consider the point process {^} of nodes generated by {gt}: each 0>i (a node) is the point of intersection of two lines from {g J . We associate with each ^ a mark kt = ttni9

where tt = ^0, nt is the polygon of the mosaic which has ^ for its utmost left vertex (such a vertex is unique with probability 1, see fig. 9.3.1). Clearly each kt is an element of IK defined by (9.3.1). We obtain a marked point process which is T2-invariant (follows from Ml 2-invariant of {gt}) and of finite intensity (follows from the observation that the number of nodes in a circle never exceeds the number of pairs of lines from {gt} which hit the circle; the latter number has finite expectation). Exact formulation Find the probability distribution p of the typical mark in { ( ^ , k()}. This formulation is a plausible interpretation of the earlier loose one because in our construction each polygon of the mosaic is represented in {^} by exactly one point (with probability 1). In this sense all polygons receive equal weights. Cases of non-equal weights will be considered in §9.6.

9.4 Solution To solve the problem of the previous section in its exact formulation we use the idea of relative Palm distribution. We have a point process on the group T 2 , namely the process of nodes

9.4 Solution

233

and a point process on the space G, namely Poisson {gJ (the two processes are strongly dependent). We mentioned in §9.3 that {rj is invariant and of finite intensity, therefore the relative Palm distribution of {gt} with respect to {ti} is well defined. We denote it as U1. The Palm distribution is concentrated on the set of realizations Jt%> = {<m G J?G : m has a node at O}. We stress that a complete description of a realization from this set includes specification of the directions of two lines through O. In fact we can assume that ^G

= [(0, n) x (0, TT)]/2 x J(G9

where [(0, n) x (0,7c)]/2 denotes the space of unordered pairs of directions. In the mosaic generated by any m e Jt% there is only one anchored polygon (see §9.3). The map realization -> its anchored polygon

(9.4.1)

is a kind of projection of Ji% on K as defined by (9.3.1). In fact the distribution p of the equally weighted typical polygon as described in §9.3 is the image of H1 under the map (9.4.1), i.e. p = projection of H1 on IK.

(9.4.2)

Proof By coincidence of the sets {^i:kieA} = {^i:timeA1}9 where AcK9 A1 = {*ne M%\ the image of m under (9.4.1) belongs to A}. Thus, (9.4.2) reduces the problem to that of the description of II x. In the next section we show that nx =AX*P

(9.4.3)

Here Ax is the distribution of a line process which can be constructed as follows. We put two lines through 0, say g' and g". The direction of g' has uniform distribution on (0, n\ and the angle \\f between g' and g" is random independent and has probability density 1/2 sin \jj on (0, n). P is the distribution of the line process {gt} with which we started, i.e. it is Poisson governed by X d#, and the * corresponds to the superposition of independent processes. Hence we come to the following stochastic construction suggested by Miles [33]: we construct the random lines g' and g" and superpose them on an independent realization of (Poisson) {#,}. In the resulting random mosaic we take its (unique) anchored polygon (doubly shaded infig.9.6.2). The distribution of the latter coincides with the distribution of the typical equally weighted polygon in the mosaic generated by {gt}9 i.e. it is p.

234

9 Poisson-generated geometrical processes

9.5 Derivation of the basic relation Let us give a derivation of (9.4.3). The problem is in the factorization of the first moment measure m1 of the marked point process {(£Ph t^1*™)}, namely in the separation of a Lebesgue measure in IR2 and in the description of the factor measure in the space of marks (normalized to become a probability measure). ^ We choose two infinitesimal domains dgx and dg2 in G whose invariant measures we denote as dgx and dg2 and an event A ~ \ ^ e assume that the union of the domains Dl9...9Ds dg2. According to (7.5.2)

covers neither dg1 nor

/\ /\ /\ Let d^ x d ^ x d^2 be^he infinitesimal domain in the space 1R2 x (0, n) x (0, n) corresponding to dgx x dg2 via the map where 9 = g1r\ g2, and the angle q>{ is the direction of gh see §2.1. The event under P on the left-hand side of (9.5.1) can be expressed in terms of the marked point process

where t^lm,\s the mark of the vertex which lies in d& and 5 = dcp1 x d(^2 x (^o1^) ^ ^ G We have (see (8.1.1) and §3.15,1) ^ x B) = EN(d0> x 5)

= X2 dg, dg2 P(A) + o( = /l 2 d^ sin \// di// dcp P(A)

(9.5.2)

(actually o(d^) = 0 because the above expectation is strictly proportional to d3>\ We have sin \j/ dij/ dq> = n o Jo

and the desired factorization receives its final form m^dgP x B) = X^d&iC1 sin \j/ d^ dcp P(A) with

9.6 Further weightings

235

Xx = 7d2. Hence the result (9.4.3). The factor n'1 sin \// d\j/ dcp (which is a probability distribution on [(0, n) x (0, TT)]/2) corresponds to Ax. We mention that k1 is the intensity of the point process {^} of vertices (nodes) generated by Poisson line process governed by A dg. The same value appears in another context in §9.11.

9.6 Further weightings In this section we briefly describe a development of the method of the previous two sections which we then use to answer some further questions concerning the random mosaic generated by Poisson line processes governed by k dg. But first we explain the main idea of weighting in terms of a general V fD ^-invariant random mosaic {raj. Let £ be some M 2 - i n v a r i a n t parameter of a polygon (such as area, perimeter length, etc). We define the distribution of a {-weighted polygon in the mosaic as follows. With each polygon n{ of the mosaic we associate a cluster of points (see §7.12, IV) cf(«) on the group M2. An essential condition EJV(Ci(
(9.6.1)

should be satisfied (the expected number of points in the whole space equals £). We put

{MJ = U c,(«>). By construction, each point of this point process is related to its 'parent' polygon nk\ f:Ms^nk.

(9.6.2)

We will refer to / as a parent map. The second assumption is that the marked point process on M 2 : {Ms, M;'nk)

(9.6.3)

(where nk = f(Ms)) should be invariant and of finite intensity. If these assumptions are satisfied, then the distribution px of the typical mark in (9.6.3) is well defined: we call this p x the distribution of {-weighted typical polygons of the mosaic. Our definition of px does not depend on the details of construction of the processes cf(co) (beyond those mentioned above). We leave the proof of this logically necessary proposition to the reader. In concrete problems it is natural to try to use the freedom in the construction of the cf(co)'s in an 'optimal' way. In the solution of the problems that follow these constructions are devised so as to obtain simpler relative Palm distributions of {gt} with respect to { M j . The expressions (9.6.4)-(9.6.6) can be derived by procedures similar to one used in §9.4, and they strongly rely on the assumption of Poisson distribution of {gt}.

236

9 Poisson-generated geometrical processes

Figure 9.6.1 The segment representing Ms begins at a vertex and follows a side of the polygon f(Ms). The latter lies in the left halfplane with respect to continuation ofM s

Figure 9.6.2 The polygon in question is shown by double shading (refers to Problem 1 below and to §9.4)

The step from these expressions to stochastic constructions of random polygons in question is by means of projections of probability measures similar to (9.4.2). Problem 1 Find the distribution of the polygon randomly chosen from the Poisson line mosaic if the weight given to each polygon is £ = the number of vertices of the polygon. The {Mt} process: we construct this as described in §9.2, Example 2. The parent map (9.6.2) is explained by fig. 9.6.1. Clearly the number of 'points' Mi which have the same parent polygons equals £, the number of vertices of the polygon. The relative Palm distributions II 2 *s given by n 2 = A2 * P, (9.6.4) where A2 is the distribution of the line process consisting of two lines through 0: one is fixed to be the x axis, the direction if/ of the other is random; \j/ has probability density 1/2 sin \j/ d^; and P is the distribution of the original (Poisson) line process {#J. The stochastic construction: We take an angular domain D of random opening \jj. In an independent realization of {#,} we take the polygon n0 which covers 0. The intersection of the two will be the polygon in question (see fig. 9.6.2).

9.6 Further weightings

237

Figure 9.6.3 The segment representing Ms has its source on and follows a side of f{Ms). The latter polygon lies in the left halfplane with respect to continuation ofM s

Figure 9.6.4 The polygon in question is shown by double shading

Problem 2 The formulation is the same as Problem 1 but with £ = perimeter length of the polygon. The {Mi} process: we construct this as described in §9.2, Example 1. The parent map (9.6.2) is explained by fig. 9.6.3. Clearly the number of 'points' M( which have the same parent polygon is random, distributed according to Poisson law with mean £. The relative Palm distribution FI3 is given by n 3 = A3*P,

(9.6.5)

where A3 is the distribution of the line process consisting of a single line, namely the x axis, and P is the distribution of the original (Poisson) line process {gt}. The stochastic construction: in a realization of the Poisson line mosaic we take the polygon n0 which covers 0. The part of n0 which lies above the x axis will be the polygon in question (see fig. 9.6.4). Problem 3 The formulation is the same as Problem 1 but with £, = area of polygon.

238

9 Poisson-generated geometrical processes

The {Mt} process: we take this to be an independent Poisson point process on the group M 2 governed by dM. The parent map (9.6.2): all segments Mt which have their source in a polygon of the mosaic have this polygon for their parent. The relative Palm distribution TL4 coincides with P, i.e. n4 = P (9.6.6) P is the distribution of the original Poisson process {gt}. This follows from the independence of {Mj and {gt}. The stochastic construction: in a realization of the Poisson line mosaic we take the polygon n0 which covers 0. It coincides (in distribution) with the polygon in question. Remark. In the latter construction we could replace {M(} by a Poisson point process on T2. The Poisson nature of {gt} was of no significance, and the result remains valid for more general random mosaics (those not necessarily generated by line processes).

9.7 Cases of infinite intensity Below we describe two examples in which attempts to follow the style of §9.2 fail because the resulting point sets on groups happen to be 'too dense' (i.e. they are no longer realizations). As shown in the next section the situation can be remedied by applying appropriate thinnings. I Let <*(<») = [Qi] be an Ml2-invariant Poisson line process on (R2 governed by X dg. We consider the triads n

s=

{Git, Gi2, Gi3}

of lines from {#J. Since with probability 1 there are no pairs of parallel lines in {gt}, each ns can be interpreted as a triangle. Modifying the description of the triangles given in §3.16 we write ns = (M s , hs, (Ts\

where Ms e M2,h e (0, oo) is the perimeter length and G e £ 2 /6 is the non-label shape of the triangle. We also put ^ s = ^\{the three lines from m which determine ns}. The random set {(M.A.^Aff 1 *.,)} represents a point process in the space M2 x (0, oo) x L2/6 x J1G.

(9.7.1)

9.7 Cases of infinite intensity

239

By an argument similar to that of §9.5 we can show that the first moment measure of this process equals X3 dM h dh mh{do)P{d*n).

(9.7.2)

Here P denotes the distribution of a Poisson line process governed by X dg (i.e. P coincides with the distribution of {g(} itself). The projection of the measure (9.7.2) on M 2 is not totally finite (because \h dh = oo). Therefore in this context the perimeter-shape distribution of a typical triangle remains undefined. The measure h dh mh(do)P(dm) in (9.7.2) is an example of what we call a relative Palm measure (if the latter is totally finite it is proportional to relative Palm distribution). II Let be a T^-invariant Poisson point process in Un of intensity X. We consider m-subsets from {Pi}m> n + 1. Each such m-subset {Pix9..., Pim} we convert into a random sequence which we denote as ({Pii9... 9 0>im})9 by applying random ordering of the points so that each of ml different ordering can be chosen with probability 1/m!. We also assume that the choices of orderings for different m-subsets are independent. Using the ideas of §4.15 we represent ({Pil9..., ^i m }) in the form ({Pil9...9Pj)

= (Aa9va9zM)

(9.7.3)

where As corresponds to the sequence of the first n + 1 points in ({Pil9..., ^im})> vs is the Lebesgue measure of the minimal convex hull of the set {&>ii9...9Pim}9 TS is the affine shape of ({0>ii9..., &>im}). We stress that for fixed {Pii9 ...9Pim} the quantities As and TS in (9.7.3) are random (they depend on random ordering of the m-set). Put *ns = ^ \ { t h e m points from m which correspond to (As9 vs9 T S )}. The set {(Aa9 vs9 TS, A;1**,)}

(9.7.4)

is a point process in the space An x (0, oo) x x nm x MUn. Acting as in §9.5 we find that the first moment measure of this process is equal to —.kmcn%m • dA vn+m~2 dv P n , m (dT)P(d*4 (9.7.5) ml where the constants cn%m generalize cm in (4.15.2) for the case where the points are in Un. Here P is the distribution of the Poisson point process {^} with which we started, and the probabilities Pnm have been considered in §§4.8,4.14 and 4.15.

240

9 Poisson-generated geometrical processes

The presence of a totally-infinite measure vn+m~2 dv shows that the relative Palm measure in this case is totally infinite. Thus the projection of the process (9.7.4) on Aw is 'too dense', and it is impossible to speak about the distribution of typical v and T values in this situation.

9.8 Thinnings yield probability distributions By applying various thinning procedures to the processes of the previous section, invariant marked processes of finite intensity on groups can be derived. One possibility is to impose restrictions on the domain of the size parameters in question. For instance, instead of {ns} (a point process in the space M2 x (0, oo) x L2/6, see I in the previous section) we can consider its thinned version {(M s , hS9 SS)}B = {ns}B,

B cz (0, oo),

(9.8.1)

where the thinning depends on a set B and is defined as follows: {ns}B = {ns: the perimeter length hs falls in B}. since MB

= M^(M2

XBXI2/6),

the first moment measure of {ns}B coincides with the restriction of A3 dgt dg2 dg3 (thefirstmoment measure of {ns}) to the set M2 x B x E2/6, i.e. it is given by PlB{h) dGl dg2 dg3 = PlB(h) dM h dh mh(da), where IB is the indicator of the set B. Since mh(L2) < oo (see §3.17) it follows that if

I

hdh = c(B) < oo

B

then the projection of the process (9.8.1) on M 2 becomes an invariant point process of finite intensity l3c(B)mh(L2/6), see (3.17.1). The perimeter-shape probability distribution for the typical triangle in (9.8.1) is then well defined and is proportional to IB(h)h dh mh(do). In words: Let the triangle process (9.8.1) be derived from an M2-myax\2int Poisson line process {g(} by a choice of B with c(B) < oo. Then for a typical triangle the variables h and o are independent and the distribution of a is proportional to mh. We have a similar situation with the process {(As9 vs, TS)} (see (9.7.3)). Taking its thinned version

9.8 Thinnings yield probability distributions

{(As,vS9zs)}B,

241

(9.8.2)

which we define by the condition vs e B, we come to the following result. Let the m-tuple process (9.8.2) be derived from a Tn-invariant Poisson {^} by means of a thinning set B for which

i

yn+m-2

B

Then the projection of the process (9.8.2) on An is of finite intensity; the affine shape T and v of the typical m-tuple are independent; the affine shape is always distributed according to Pnm(dz). Now let us consider other types of thinnings which do not depend on arbitrary truncations of the parameter space. From the process {ns} we delete all triangles which fail to satisfy the condition the interior of ns is intersected by / lines from {gt}. What remains after this thinning operation we represent by {*,}, = {(A*,,fc.,*.)},(9.8.3) (If / = 0 this is the collection of all triangular tiles of the mosaic formed by Our process {7rs}, can be obtained from the marked point process (9.7.1) by two operations: (a) restricting the process (9.7.1) to the set ax = {{n,m)\l lines from m hit the triangle n}\ and (b) projecting the resulting process on the space of triangles. In terms of the first moment measures, operation (a) corresponds to a passage from (9.7.2) to Iai(n, *n)P dM h dh mh{da)P{d^n\ Operation (b) corresponds to the integration of the above by P(dwi). We get A3 dM h dh mh{da)

Iai(n, *n)P(d*n)

\ dh mh(do)(kh)l(V)~l exp( — kh\ which is the first moment measure of {ns}t. The expression 1

jj^y

exp(-A*)A dh mh(da)

(9.8.4)

(proportional to the previous expression) is a probability measure in the space (0, oo) x E 2 /6. This means that (9.8.4) presents the perimeter-shape distribution of the typical triangle in {rcj,. We stress that perimeter and shape are

242

9 Poisson-generated geometrical processes

independent and the shape distribution does not depend on /. Lastly, the intensity of the projection {Mt} of (9.8.3) on M2 equals (see (3.17.1)) O

11

Let us consider a similar thinning of the process (9.7.4). In the space AM x (0, oo) x xn m x J R n we take the set bx = {(A, v, T, *n): there are / points from <m in the domain whose volume we denoted by v}. We define the thinned process to be {(As9 vs, TS9 AJ1<ms)}l = btn {(AS9 vs, TS, A'1^}.

(9.8.5)

The first moment measure of this process is the restriction of the measure (9.7.5) on bt. The projection of this restriction on the space An x (0, oo) x xnm is —-c n ml

m

&A(AV)1{V)~1

exp(-Av)vn+m~2

dv Pn m(dx).

(9.8.6)

Because projections of first moment measures correspond to projections of point processes, we conclude that (9.8.6) coincides with the first moment measure of the process {(As9 vs, xs)}i = {those m-tuples from {^} whose minimal convex hull contains, apart from the points of the m-tuple itself, / points from ^ } . In this way we come to the following conclusion. Given a Tn-invariant Poisson point process {^} in R", we construct the thinned process of m-tuples {{9h9...,

&ij}i = {(As9 vS9 xs)}t.

(9.8.7)

The distribution of typical mark (v, x) is well defined for this: v and x are independent, v is gamma-distributed, the distribution of x does not depend on I and is always given by Pnm. The first moment measure of the projection of the point process (9.8.7) on the space (group) An equals

In particular (see (4.8.2) and (4.14.2)) this equals 5(/ + 1)(/ + 2)(/ + 3)(Z + 4) J A 6/ dA

m the case n = 2, m = 4

and 189

6

—j- Y\ (I + i) dA

in the case n = 3, m = 5.

Projections of these measures on Un are not locally-finite. In order to obtain locally-finite projection measures, thinnings in the space of'Euclidean' shapes are necessary. An example of such thinning can be found in the next section.

9.9 Simplices in the Poisson point processes in Un

243

9.9 Simplices in the Poisson point processes in Rn Here again {^} denotes a Poisson point process in Un governed by the Lebesgue measure X d^, and P denotes its distribution. Results concerning the shape of the simplices generated by {^} can be obtained by reducing the problem to marked point processes on the Euclidean group Mn. We describe a simplex 9 (a set of n + 1 points) as

9 = (M, v, o\ where M e M n determines the position of 0, v > 0 is the volume of 9, and a is its shape with no label. In the space M2 x (0, oo) x £n/(n 4- 1)! x JtUn we consider the point process. {(Ms, v,9 <JS9 M ; 1 ^ ) } ,

(9.9.1)

where each (Ms, i?s, crs) corresponds to a simplex 0S with vertices from {^} and ^ s = {^.}\the vertices of the simplex 9S. Applying the reasoning of §9.5, the first moment measure of (9.9.1) is found to be kn+1 dM v"'1 dv mv(do)P{d™\

(9.9.2)

Here mv(da) is a measure in the space En/(n + 1)! of simplex shapes in Un determined by the decomposition A»1 . . . d0>n+1 = dM v"-1 dv mv(d(7%

where d ^ are Lebesgue measure elements. Explicit expressions for mv have been obtained for the cases n = 2, 3 in §4.6 and §4.12. Clearly (9.9.2) can be considered as another instance of calculation of the relative Palm measure (which is now totally infinite). The value mv(Ln) of the whole space of shapes is infinite (consult §4.6 and §4.12). We come to the conclusion that, in order to obtain a marked point process on Mn of finite intensity, thinning of (9.9.1) by imposing the conditions (a) vs belongs to a fixed bounded set or (a') there are exactly / points from {^} within 9S. is not enough. In order to obtain finite intensity, it is sufficient to supplement (a) (or (a')) by an additional condition of the type (b) os belongs to a fixed subset of the space of shapes, whose my-measure is finite. In the resulting thinned processes {M s , i;s, os}ayb

or

{M s , vS9 os}a,,b

the shape distribution of a typical simplex will be proportional to the restriction of the measure mv to the set mentioned in (b). In this sense (which is usual in

244

9 Poisson-generated geometrical processes

classical geometrical probability) the distribution in question is governed by mv . Perhaps more convenient is another type of thinning which we describe for n = 2. Assume that our Poisson {^} is superposed by an independent line process {gt} which is also Poisson and governed by kx Ag. We consider all triangles 0s = 0>{i gp^ gp^ which are hit by exactly / lines from {#J. We denote this set of triangles by {8s}t. Note that {9s}l=0 is just the set of triangles 0S, each of which lies within a tile of the mosaic generated by {gt}. Now we describe a triangle by the parameters 6 = (M, K o\ where M and o are the same as before, and h is the perimeter length of 6. In the space M2 x (0, oo) x £ 2 /6 we consider the point process {(Ms, hs, 9S)} = {ex,},.

(9.9.3)

Its intensity measure can be easily seen to be (see §3.16, III) P AM h3 Ah v^Aa^hyiliy1

exp(-/l1/i).

(9.9.4)

By integration we find that the projection of the process (9.9.3) o n M 2 has the first moment measure (see §6.11)

From (9.9.4) we conclude that the perimeter length h and the shape a of a typical triangle in {0S}1 are independent; h is gamma-distributed, and the law of o e L 2 /6 for every / is (126/7i) • vh.

9.10 Voronoi mosaics Similar ideas can be applied in the study of the so-called Voronoi mosaics generated by T2-invariant Poisson processes on U2. First we recall their definition. Given a realization set m, of points in R2, we associate a convex polygon ni with each point ^{ e <m\ by definition nt consists of all points of U2 which lie closer to ^ than to any other point of m. An equivalent construction of TT, is the following. Let us denote by gtj the line which is perpendicular to the ^ ^ - segment in its midpoint (the midperpendicular), and let {Gij]i = {9ij • i is fixed, ^ e m\. If htj denotes the halfplane bounded by gtj which contains the point 9{ then n{ = f) hij.

(9.10.1)

j

Now let <m — *n(a>) = ffi} be random, namely a T2-invariant point process of

9.10 Voronoi mosaics

245

finite intensity. Then the above construction yields a polygon process {TCJ which in fact is a random mosaic (the Voronoi mosaic for m(co)). It is natural to consider the marked point process m , tt{Qv}t)}> w h e r e *t=^° (9-10-2) (marks are line processes). It is T2-invariant and of finite intensity. The distribution of its typical mark will be that of a stochastic line process; we denote it by p. A line process with distribution p can be constructed in two steps: (a) take a point process {Qs} whose distribution coincides with the Palm distribution of {^}; (b) take the set {gOs} of midperpendicular lines for all pairs 0, Qs. The proof that {gOs} is the desired line process follows directly from the definition in §8.1. The line process {gOs} generates a random mosaic, and in it the polygon which covers O is well defined. The latter polygon coincides in distribution with the typical equally weighted polygon in the Voronoi mosaic {TTJ generated by The distribution p can be further specified if we assume that {^} is Poisson governed by k • L 2 . In this case outside 0, the process {Qt} will also be Poisson governed by kL2. Representing the points Q, in the usual polar coordinates we write Clearly {(ri9 (p{)} will be a Poisson point process in the space (0, oo) x S x governed by the measure kr dr dcp. Corresponding to that, the line process be Poisson governed by 4kp dp dcp (see §2.1). We conclude that is Poisson governed by k • L 2 then the typical equally weighted polygon in the Voronoi mosaic {TCJ coincides in distribution with the polygon which covers O in the mosaic generated by Poisson line process governed by the measure 4kp dp dcp = Akp dg. If {^} is Poisson, we can also solve the problem of the probabilistic description of the shape of a typical vertex in the corresponding Voronoi mosaic. There is an elementary geometrical characterization of the collection of vertices in terms of the set {^}. Namely, a point v e IR2 is a vertex of {nt} if and only if in {^} there is a triad 0>h 0>p 0>s for which (a) v is the center of the circle through ^ , ^ , 0>s\ (b) the interior of this circle does not contain any points from We denote by {vr} the set of vertices of Voronoi {TTJ. If {^} is Poisson (and this we now assume), there are no quadruples of points in {^} which lie on a

246

9 Poisson-generated geometrical processes

Figure 9.10.1

circle with probability 1; it follows from (a) and (b) that with probability 1 each vr is of fork type (seefig.9.10.1). The shape of vr is determined by two angles between edges emanating from vr and is completely determined by the shape of the triangle ^h 0>j9 0>s for which vr is the circumcircle (see fig. 9.10.1). Therefore it is enough to describe the shape of the typical triangle in the triangle process {^,^,^,}(b),

(9.10.3)

(which is the set of triads from {^} which satisfy the above condition (b). The problem can be solved by applying the machinery of the previous sections in the following steps: (1) we consider triads {0>i9 ^p 0>s) of points from {^}. We represent the rth triad by means of parameters MreM2 which determine the position of the triad; Ar = the area of the circle through the points of the rth triad and the triangular shape cr e £ 2 /6; £ 2 /6 is the space of triangular shapes without labels. (2) calculation of the first moment measure of the point process {(Mr, Ar,
(9.10.4)

2

Using (3.17.2) we find the projection of this measure on U to be 22 d^. This is the first moment measure of the process of vertices of our Voronoi mosaic. We also gather from (9.10.4) that the shape of the typical vertex is

9.11 Mean values for random polygons

247

distributed according to the density n sin ax sin a 2 sin a 3 d ^ d£ 2 , in the space £ 2 /6. The value 2X for the intensity of the vertex process agrees with the value 6 for the mean number of vertices in a typical polygon in Voronoi mosaic obtained by a different method in the next section.

9.11 Mean values for random polygons The stochastic constructions of random polygons of §§9.4,9.6 and 9.10 can be applied in Monte-Carlo calculations. They also permit exact evaluation of the expectations of various random variables depending on these polygons. We show this in the examples of A - the area of the polygon; H - its perimeter length; n - the number of its vertices. Let us start with the derivation of certain integral representations for expectations EA, EH and En for n0 - random polygon which covers the origin O in a Poisson line process on IR2 governed by a measure f(g) dg. We will assume that the density f(g) is continuous; in our stochastic constructions we met two cases: / = X and / = 4Xp. We consider random measures A(B) = area of n0 n B, B a U2 is a Borel set; H(B) = \dnonB\, the total length of dn0 n B; n(B) = the number of vertices of n0 within B; y(B) = £ \gt n B\9 where summation is over all lines gt from the realization of the line process; N(B) = number of gt n gj points in B. Clearly A = A(U2)=

\A(dQ),

H = H(U2) =

H{dQ\

n = n(U2)=

[n(dQ\

248

9 Poisson-generated geometrical processes

The application of averaging and the Fubini theorem yields

EA = I EA(dQ),

-I

EH=

EiJ(dg),

(9.11.1)

En = I En(dQ),

•J

where E,4(dg), EH(dQ) and En(dg) are the corresponding averaged measures. In our case these have densities with respect to dQ. The purpose now is to calculate these densities. We note that for areas dQ which shrink down to a point g we have (up to summands which are o(dQ) in the average) = Io(OQ)'dQ, = I0(OQ)X(dQ),

(9.11.2)

= I0(OQ)'N(dQ)9 where I0(OQ) — 1 if the segment OQ is not intersected by the lines of the process which avoid hitting dQ, and zero otherwise. The random variable I0(OQ) is independent of either dQ, J£?(dQ) or N(dQ) (this follows from the independence properties of the Poisson line process, see (7.5.2)). Taking expectations of (9.11.2) we find

EH(dQ) = P flO®\ E&(dQ\

(9.11.3)

Because the process is Poisson

'(T)--'1-.L™* 1 = exp(\ he

(9114)

where as usual [Og] is the set of lines which hit OQ. The main term of E<£{dQ) is contributed by the event in which only one line hits dg, i.e.

•I where / is the length of the chord g n dQ. Using considerations suggested by fig. 6.3.1 we find

ff

f(g)d
)

where [ g ] denotes the bundle of lines through g.

(9.11.5)

9.11 Mean values for random polygons

249

Lastly, using the notation of §3.15,1, f(g1)f(g2)

sin * dxjj dcp) dQ

(9.11.6)

(the factor \ appears here because the two lines that hit Q should be considered as unordered). Formulae (9.11.1) and (9.11.3)—(9.11.6) reduce the problem to performing integrations over a plane. The simplest is the case in which / = L In this case,

p

\ cTJ=exp(-2Ar)

(we use polar coordinates Q = (r, 0)), and

Therefore

•-JJ-*-

exp(-2Ar)rdr d0 = n'{2k2)~\

EA3 =

exp(-2/lr) r dr d0 = n2(2X)-\

(9.11.7)

2 = nX2 \\ exp(-2Ar)r dr d0 = -TI T .

These mean values correspond to the random polygon of Problem 3 in §9.6. The same values can also be found in [1], section 6.3. The integrations in the case/(#) = 4Ap are somewhat more complicated but still tractable. Below we present the results for this case. For a Poisson line process governed by 4Ap d#, (9.11.4)—(9.11.6) give

since f(g) dg = 4/1 [' dx | x sin 2 ij/dij/=

I JIOQ]

Jo

Anr2;

Jo

f* r sin

E<£(dQ) = M• dQ1

Jo Jo EN{dQ) = dQ U2r2

J Jo

= dQ 8A 2 r 2

Jo

= 6;ryl2r2 dQ.

sin (p dcp d sin cp

J Jo

\sm(ij/ +
sin cp dcp-[ I — — cp) cos G) + sin cp I 2

VV

/

/

250

9 Poisson-generated geometrical processes

Formulae (9.11.3) and (9.11.1) now yield f00 EAV = 2n \ r exp(-Xnr2)

Jo

f00

EHV=U2n

Jo

dr = A"1;

r2 exp(- Xnr2) dr =

2

EnV = 6nX • 2n \

r3 exp(-Xnr2)

dr = 6.

Jo According to the results of the previous section these mean values refer to the typical polygon in the Voronoi mosaic generated by Poisson point process on U2 governed by X d£P. The values (9.11.7) can be used for direct calculation of similar expectations for the random polygon of Problem 2, §9.6 (we call them EA2, EH2 and En2): EA n EA,l2 = —

= • "l"~4F'

EH EH2 = - r - plus the average length of the side on the Ox line

En2 = — + 2 = 2 + By slightly modifying the derivation which led to (9.11.7) we can obtain the values of EA, EH and En, for both the random polygons of §9.4 and of Problem 1, §9.6. The counterpart of (9.11.2) now has the form (see fig. 9.6.2)

H(dQ) = ID(Q)Io(OQ)X{dQ), n(dQ) = ID(Q)I0(OQ)N(dQ). Except for A, the total values of these random measures do not represent the desired random quantities. We have in fact = I A(dQ),

-J

H(dQ) + \OA\ + \OB\9

n=

n(dQ) + 3. j

Let us consider the representation for A. Because of the independence of ID, the result of averaging will be

9.11 Mean values for random polygons

251

fao rn/2

EA=

\

p(0) exp( - 2lr)r dr dO

f /2 p(O)dO, -n/2

where p(0) is the probability that the point Q = (r, 6) will fall in the random angular domain D. To calculate the last integral we note that rn/2

rn/2

p(6) dO= \ J -7T/2

rn/2

d0 EID(0) = E

J -7T/2

ID(0) d0 = E\D\9 J -Jt/2

where |D| denotes the (random) opening of the angular domain D. The reader may easily verify that in the case of §9.4 \D\ has a probability density function 7C"1(TC — a) sin a da,

0 < a < 7c;

hence for equally weighted typical polygon in Poisson {gt} _ rn 4 E\D\ = 7t M a(7r — a) sin a da = —, Jo

7C

and

EA = (U2y1'4n-1 =7T 1 /T 2 . With all the above calculations wefindeasily EH = T d ^ T 1 -47T1 + /T1 = 2 r \ En = T d 2 ^ 2 ) " 1 -47T-1 + 3 = 4. These values can be found in [2], p. 57. In the case of Problem 1 in §9.6, \D\ has a probability density function - sin a da,

0 < a < 1.

This yields the following values (the index 1 stands for Problem 1):

En, = — + 3.

10 Sections through planar geometrical processes

We now consider Ml 2 -invariant geometrical processes of a general nature (we do not assume they are Poisson or Poisson generated), namely line processes, random mosaics and Boolean models for disc processes. Our interest lies in intersection processes induced on a 'test line' (the x axis, say). The essential feature of the analysis is that it includes the angles under which the intersections occur. In other words, we study the induced one-dimensional marked point or segment processes with angular marks. The first three sections are partly preparational; here we develop the ideas of chapters 8 and 9. The main results of this chapter are obtained in the latter five sections where a principally new tool is introduced: the averaging of combinatorial decompositions of chapter 5. The basic relations which we derive (such as (10.4.11)) are of stereological significance. They permit the conclusion that M2-in variance implies strong restrictions on the possible distributions of intersection processes. In particular they imply that additional renewal-type properties alone may determine the functional form (exponentiality) of distribution functions of certain length variables.

10.1 Palm distribution of line processes on U2 Let {gt} be a line process on U2 (a point process in G; see §7.5, Poisson nature is no longer assumed), and let P be its distribution. We assume {gt} to be M 2-invariant and of finite intensity. The latter condition means that for every compact B c G EPN(B, m) < oo,

m = {gt}.

Because this expectation is a measure in G and inherits the property of y 2 " m v a r i a n c e fr°m P* w e conclude that EPN(B9»*) = A-ti(B)9 (10.1.1) where \i is the M]2-invariant measure on G described in §3.6. We stress that

10.1 Palm distribution of line processes on U2

253

(10.1.1) follows from the uniqueness of such \i. X < oo is called the intensity of {gt}. First using {g(} we construct a point process {Ms} on the group M 2 , namely that described in §9.2, Example 1. By construction, the processes **!(a>) = {Ms},

^ 2 (co) = {gt}

are jointly M 2 -invariant. Hence the relative Palm distribution Il{Ms} of {gt} is well defined. We call it the Palm distribution of {#J and denote it simply byEL Let us mention from the beginning that IT{the x axis belongs to a realization} = 1, and it can be convenient to think of II as the conditional distribution of {g(} given that one line from {gt} coincides with the x axis. In (9.1.3) as applied to our case we put B1 = {MeM2:M0eb(r90)}9 A = B2 = {kt lines from {gt} hit dh i = 1 , . . . , s}, where b(r, 0) is the disc centered at O of radius r, and 5l9...,5s segments fixed on the plane. We obtain U(A) = (n2r22)-1E

£

IA(M^).

are some

(10.1.2)

MseBi

(the intensity of {Ms} equals Xjl). To calculate the expectation in (10.1.2) we may first take the lines gt e m, as fixed and average with respect to the positions of the segments Ms on these lines. After that the result should be averaged with respect to P, the distribution of {#,}. Applying the first step, we obtain an expression proportional to the righthand side of (6.8.10), i.e.

X

I

9i hitss b(r,O) J xx((9i)

A(M£lttm)

+ IA(Mrltt*n)) dt.

(10.1.3)

On the other hand, we can obtain the same expression by applying the expectation EP to both sides of (6.8.10) and acting as we did in §7.15. We find in this way that U(A) = lim (XI2)-1 d'P(AnB),

(10.1.4)

l-"O

where (seefig.10.1.1)

V

O Figure 10.1.1 d = x2-xul=

x2 |vt| = |v2|

2

254

10 Sections through planar geometrical processes

B = {a line from {g{} hits both v1 and v 2 }.

(10.1.5)

P(B) = MTH2 + o(l2).

(10.1.6)

In particular, This permits the reinterpretation of H(A) as a limit of conditional probability: .

(10.1.7)

The right-hand side of (6.8.10) did not depend on the locations xx and x 2 along the x axis of the segments vx and v2. Consequently, for any fixed A the limit in (10.1.4) does not depend on xx and x 2 . This implies that the Palm distribution II is always invariant with respect to shifts parallel to the x axis. Proof Using the same notation as before,

P(A nB) = P(tA n tB) = P(tA n B) tB={tm\m€ B) for every shift t parallel to the x axis. Dividing both sides by Id'112 and taking the limit, we obtain U(A) = U{tA).

(10.1.8)

Conditioning by the event (10.1.5) is not the only possible method. For instance in (10.1.7) we could take B = {a line from {gt} hits an interval (x, x + dx) on the x axis under the angle within (ij/, \jj 4- d^)}. The corresponding limit will exist for any ij/ ^ 0 or n and will produce a rotated version of II, namely I1X,V(A) = ll{Mx^A\

(10.1.9)

where MXtW denotes the rotation around the point x by the angle ij/. The Palm distributions for doubly stochastic Poisson line processes can be easily described. We consider two examples (proofs can be obtained by (10.1.7) and are left to the reader). Randomly rotated T2-invariant Poisson line processes (These processes have already been described in §7.12, II) If the process {#J is obtained by random rotation from a Poisson line process governed by the measure d/^ = f((p) dcp dp then the Palm distribution of {#J will be doubly stochastic Poisson governed by random measure f(q> - oc) Aq> dp, where a is a random angle with distribution

10.2 Palm formulae for line processes

255

Here X is the intensity of {gt}, the fact that (nX)'1 is a norming constant follows from (2.10.1). Mixtures of M2-invariant Poisson line processes Let {gt} be a doubly stochastic Poisson line process whose governing measure with probability 1 is proportional to dg but with random proportionality coefficient X^ (here co denotes an elementary event rather than a direction). Let f(x) dx be the probability density of X^. The Palm distribution of {gt} will be doubly stochastic Poisson governed by X'^ dg, where %!w is a random variable with probability density

(I

xf(x) dx

10.2 Palm formulae for line processes The distribution P of any M 2 " m v a r iant line process {gt} of finite intensity X is uniquely determined by X and the corresponding Palm distribution II. A proof suggested in [42] is by reconstruction of the probabilities (10.2.1) in terms of X and II. Above 8U ..., 8S are intervals not necessarily on a line; [(5] is the set of lines hitting 5. We assume that the line which contains 6 does not belong to Buffon set [<5]. This convention will be important when we consider Palm probabilities of the events as in (10.2.1). This approach is based on the fact that the probabilities of (10.2.1) type determine the distribution of a line process uniquely, i.e. the sets [(5] can in this respect replace the 'shields' described in §7.5. This of course needs a separate proof, which we leave to the reader. For simplicity we consider the probability

assuming that the interval 5 lies on the x axis. The point process {xj of intersections of the lines {#,} with the x axis will be of intensity 2X. Applying the theory of §8.7 we find that {t) 2X[-l{t)-nk{i)\

(10.2.2)

where nk(t) is the conditional probability to have k intersections on S given

256

10 Sections through planar geometrical processes

Figure 10.2.1 S = (O, t)

that an intersection has occurred at an end of d (see fig. 10.2.1), in the sense of Palm distribution of a point process on a line. How can the probabilities nk(t) be expressed via the Palm distribution II of the line process The conditional probability of the event I

1 given that there is an

\ k ) intersection at x and the intersection angle at this point equals ij/ is Hx v ( see (10.1.9); the random angle \\t has \ sin \// dij/ density (see §7.15, IV). Therefore

nk(t) = l- P n 0 .v

sin +

ty.

(10.2.3)

The two preceding equations, together with (10.1.9), solve the problem of reconstruction of P

k The probabilities (10.2.1) can be recovered by the method used in §8.8 (no additional difficulty stems from the possibly non-collinear position of the intervals d^. One can prove similarly that the only solutions of the equation

n = A*P are provided by Poisson line processes. Here A{m consists of a single line, the x axis} = 1.

10.3 Second order line processes Second order line processes offer a number of analytical advantages, and in fact the first attempt to develop a theory of general (non-Poisson) line processes by Davidson [43] was restricted to that case. We too will assume the second order property throughout this and the following sections. I Let P be the distribution of an M 2 ~ m v a r iant line process {gt} of finite intensity and let n be its Palm distribution. If 3l9 <m)N(B2, m) < oo for compact

BUB2<^G>

10.3 Second order line processes

257

i.e. if {gt} happens to be of second order then EnN(B, m) < oo

for compact

B a G.

Proof Since

using linear properties of (10.1.3) we find that this quantity is proportional to

Xi

where Xt are the chords of ft(r, 0) generated by {g J (h(r, 0) is the circle of radius r centered at 0). We can take B = [fe(r, O)]. Then the expression under the sign of EP does not exceed 4rN(b(r, O), ™)N(b(2r, O\ m). By assumption, EP of this product is finite. II Under second order assumption it is easy to prove different continuity properties of the Palm distribution. For instance, let Ix,..., / s be a sequence of bounded intervals on the x axis, and let kl9..., ks be a sequence of nonnegative intergers. Then for every x $ (J /,•

'
K)'

Clearly the above is equivalent to limn

V ki

,...,,

K)

=n

,...,.

\K

K)

.

where the intervals Il9..., 7S lie on the line x, \J/ and are shown in fig. 10.3.1. Let us prove (10.3.1) assuming we have only one interval I1. We have the inclusion |, A)], ^ ) > 0} U { - : iV([6(|v2|, B)], «.) > 0} (the segments vl and v2 are shown in fig. 10.3.1). Each of the latter two sets is monotone decreasing with \j/. Therefore it is enough to show that JC, i// line

A Figure

10.3.1

/j

\xA\ =

B \xC\...

I2

258

10 Sections through planar geometrical processes

II {there is a line through A or B in a non-zero direction} = 0. This probability equals zero since, by the result of I above, where 0 < X' < oo, and N* denotes the number of lines having non-zero direction. The case of general s in (10.3.1) yields a similar analysis. Ill We consider the one-dimensional marked point process description of line processes where xt is the point of intersection of gt with the x axis and ^ is the corresponding intersection angle (see §7.12, II). We will be interested in the second-order fMl ^-invariant line processes which satisfy the condition (J) the intersection point process { x j and the sequence of angles {^} are independent (independence of the angles within {^} is not required). This class of line processes is not empty for it contains mixtures of M 2 ~ m v a r i a n t Poisson line processes, see §10.1. We show now that (./) implies strong restrictions on the structure of the process { x j . For the events A = (^,..., ^

),

/, lie on the x axis,

(10.3.2)

condition (
(10.3.3)

identically for every if/ and for every x outside the union of Il9.. , Is. One of the consequences of (10.3.3) is that the quantity

=\ P (which has the interpretation of the conditional probability of A under the condition that x is a point in {xj) does not depend on x: UX(A) = II(,4) for all x $ (J It. In particular, if we take s = 1 in (10.3.2), (10.2.2) and (10.3.3) may be written as (10.3.4) We say that a Tx-invariant finite intensity process { x j c U has the Palmmixing property lim UX(A) = P(A)

for every A as in (10.3.2)

10.3 Second order line processes

259

If we now assume that our process {xj is Palm-mixing then under our other assumption it follows that

Substituting this value into (10.3.4) and solving the resulting equations system (see §8.11) we conclude that

Similar reasoning in conjunction with the integration method of §8.8 yields ,

ks )

V

k{\

i.e. if { x j is Palm-mixing and (J) holds, then { x j is Poisson governed by 2XLl. We stress that there is no difficulty in constructing examples of second order Jx -invariant line processes {9i} = {(x,, *,)} for which both properties {J) and of Palm-mixing of { x j exist. For instance, if {xj is a Tx-invariant finite intensity renewal (see §8.9) with non-lattice-type distribution F possessing second moment, then {xj is Palm-mixing (this is in fact one of the main results of the renewal theory [54]). To satisfy {J) for {^} we can take any stationary sequence of angles which is independent of { x j . It follows that within this class, say, M2-invariance of corresponding {gt} will necessarily imply exponentiality of F. IV A family of Palm-type distributions For second order M 2 -invariant line processes {#J, which with probability 1 do not possess pairs of parallel lines, a limiting procedure in a sense dual to (10.1.7) leads to a family of Palm-type distributions of {#J which we denote as TLXlX2With fig. 10.1.1 we now associate the event B = {<m e JiG : m contains exactly one pair of lines which belongs to B*}, (10.3.5) where B* = {(0i, Gi)' 9i * 9i, 0i hits v1? g2 hits v2}. We denote by 6t the random angle between v, and the line hitting v£. / = 1,2 (these angles are well defined if B takes place). We choose an event A cz M^ as in §10.1 and two arcs Jx a (0, n) and J2 <= (0, n). We denote by [ J J the event ©, e Jt. Let P be the distribution of a line process of the above-mentioned class, and let m2 be its second moment measure. Then

260

10 Sections through planar geometrical processes

P(B) = m2(B*) + o{l2) = cl2 + o(l2).

(10.3.6)

Here / = |vt | = |v2| tends to zero, and the constant c is given by the integral (7.14.4). The limit of the conditional probability 29

,

p(B)

0

exists and for each x x and x2 determines a probability distribution in the space

Let us first give a proof for (10.3.6). We denote by N2(B*, m) the number of pairs of lines from m which belong to B* so that (see §7.14) m2(B*) = EPN2(B*, m). It is not difficult to establish both the existence and coincidence of the limits 1-22 lim/" I

Is(M™)dM

,0)X(0,2TT) Jb(r,C

= lim r 2 l->0

N2(B*, Mm) dM = a(m)

(10.3.8)

Jb(r, 0)X(0,2TT)

for the set of realizations m e J?G of probability P = 1. For / < 1 both ratios under the limit signs do not exceed

Z gitgjhitb(r+d+ltO)

(where a 0 is the angle between gh g} e m9 see §6.5). But as we already mentioned in §7.15, V) this upper bound may have infinite expectation EP. Therefore an attempt to directly use Lebesgue's bounded convergence theorem fails. To remedy the situation, instead of B* and B we consider the sets B* = {(#!, g2) e B* : the angle a 12 between QX and g2 belongs to (s, n — e)}, £ >0

and Be =

{™eJ?G:N2(B?,*n)=l}.

A limit similar to (10.3.8) exists with probability 1: f lim r 2 AT2(£e*, M*n) d M = aE(m\ 1-+0

Jb(r,0)x(0,2K)

and the upper bound

I gt,gjhitb(r+d+l,O) E
of the expression under the limit sign has a finite expectation

10.3 Second order line processes

261

L,^hitfc(r+d+i.o) ( y - a i 2 ) c o t a 1 2 + 1 \m2(dg1dg2 (the integrand is bounded on the integration set, the latter set is bounded). By Lebesgue's theorem we conclude that lim r2m2(B*) =

(^-inY^aJ^n).

1-0

Observing that ae{m) is monotone increasing as s 10 we conclude the existence of the double limit: lim lim r2m2(B*) = (nr2 • 2n)'lEPa(m). eiO l->0

From the results of §6.5 we conclude that ™2(B?) = I2 \

( y - a 12 ) cot a 12 + 1 m(da 12 )

J7r-£>a12>£ L \ Z

/

J

and the previous double limit is directly calculated to be c = l~2m2(B*) = r2EPN2(B*) = {nr2'2nTlEPa{™). Now we apply EP to both sides of (10.3.8). By the Fatou lemma lim sup r2P(B) ^ lim inf l~2P(B) > (nr2 • 27i)-%aM = c.

(10.3.9)

On the other hand, since lv(Mm) ^ N2(B*, we have lim inf l~2P(B) ^ lim sup l~2P(B) < lim r2EPN2(B*, *n) = c. (10.3.10) The inequalities (10.3.9) and (10.3.10) together imply (10.3.6). The above argument applies with minimal changes if we replace the event B by [ J x ] n [ J 2 ] n AnB, and thus the existence of the limit lim r 2 P ( [ J J n [ J 2 ] n A n B) 1-+0

can be established. From this and (10.3.6) follows (10.3.7) and the complete statement. Remark In view of the results on pairs of segments and pairs of disc mentioned in §6.5, a natural question arises: do propositions similar to the above exist for fy2-invariant second order random segment processes (especially for random mosaics) or random disc processes? The answer is that under additional assumptions of the existence of the densities as interpreted in the examples of §7.14 the natural reformulations of (10.3.6) and (10.3.7) can be proved without substantial difficulties. At present it is not clear whether the above-mentioned additional assumptions can be removed; therefore we adopt them in our treatment of random mosaics and disc processes in §§10.6 and

262

10 Sections through planar geometrical processes

§10.7. To stress the common nature of the conditioning event (each of the vertical windows is hit and the hits are produced by distinct elements of the realization) we preserve in these sections the notation flXlX2 for the corresponding limiting (/ -> 0) conditional distributions.

10.4 Averaging a combinatorial decomposition In this section we again consider general line processes {#J which we assume to be M 2 -invariant, of second order and possessing with probability 1 no pairs of parallel lines. The aim is to average a combinatorial decomposition in the style of (5.4.1) and (5.4.2) where the inherent set of'needles' depends on {#J, i.e. is random. The resulting quantities will depend on a small parameter / and we will be putting down their main terms which are of order I2. I The combinatorial decomposition We consider the segments vx and v2 shown in fig. 10.4.1 as the sides of a rectangle R, the vertices of which we denote by Qi9 i = 1, 2, 3, 4. Let Xi = Qi^R

(10.4.1)

be the random chords generated by the lines of {gt}; we denote by 0>{ the endpoints of these chords. With probability 1 the set {^} is finite. The algorithm (5.1.3)—(5.1.2) can be easily applied to find the combinatorial decomposition for the set C n B e K { 6 J u {^}X where C = {g e G : g hits both vx and v2} = [ v 1 ] n [v 2 ], B = {g e G : g does not hit any of the chords &}. There is a slight complication in that with positive probability there can be collinear triads of endpoints ^ . We overcome this by replacing the sides of R by outward circular arcs, writing the combinatorial decomposition for naturally redefined CnB and taking the limit whilst letting the curvature of the arcs tend to zero. The final result is as follows: H(B n C ) = fl1+fl2 + fl3 + a 4 + a 5 ,

(10.4.3)

where

V

QT

'

^ 4

Figure 10.4.1 \QU Q4\ = t, W = |v 2 | = /

2

10.4 Averaging a combinatorial decomposition

263

= -4>«2i. Q*)\Qi, QA\ - hiQi. 63)162, Gsl,

= io(Qu Q3)\Qu Q3\ + h{Qi> Q*)\Qi, CJ, = "2 E

\Xi\hilil

In these equations: // is the M 2 -invariant measure on G;\0*l9&2\ denotes the Euclidean distance between the points ^l9^2l writing Xt e C, ^ ^ e C o r g £ ^ e C means that the corresponding line should belong to C; in the expression for a4 the sum is extended over pairs &x ^ which fail to be endpoints to a xsl I0(S) = 1 0

if no closed chord Xj hits the interior of the segment 3 (if S = Xi then we assume that j ^ i); otherwise

Lastly, the events d and s correspond to the classification introduced in §5.4 and are shown graphically in fig. 10.4.2.

(b)

(c)

(d)

Figure 10.4.2 (a) ^ ^ - is of d-type (x, and Xj lie in different halfplanes with respect to the tPi&j line), (b) ^ ^ - is of 5-type (xi and Xj lie *n one halfplane with respect to the ^ line), (c) Qi&jis of ^-type (vt and Xj lie in different halfplanes with respect to 9j). (d) Qi&jis of s-type (vj and Xj lie in one halfplane with respect to Q^j).

264

10 Sections through planar geometrical processes

II Averaging in the case of infinitesimal / The aim now is to calculate EP £ ai9 see (10.4.3). The length / of both vx and v2 is assumed to be infinitesimal. We have (see the notation of §10.2)

= EEPP ff IIBB(g) (g) dg dg == fI P(B) dg Epfi(B n C) = =

I

Po(\R n 0l) ^ = r ^ o W / 2 + o(/2);

(10.4.4)

in the last step we used (3.7.4). Then EP(ax + a2) = 2(t2 + / 2 ) 1/2 p 0 [(^ 2 + / 2 ) 1/2 ] - 2tpo(t) + o(' 2 ).

^

(10.4.5)

To evaluate EPa3 we note that, according to (10.1.6), the contribution of the cases where the sum consists of more than one summand is o(l2). Therefore according to (10.1.7) EPa3 = -2tP(

An (

1

) ) + o(/ 2 ) = -2A/27c0(t) + o(/2),

(10.4.6)

where

The expectation E P a 4 is easily evaluated in terms of the distribution ftQiQ4 because the expected contribution of the cases where the sum consists of more than one summand is o(l2) (see §10.3, IV). With B as in (10.3.5) and the events {d} and {s} corresponding, respectively, to parts (a) and (b) of fig. 10.4.2, we have =

a^P(dm) + JBn{d] Bn{d]

a4

4- o(l2)

JBn{s]

= cl2t[UQiQ4(A n {d}) - nQlQ4(A n {s})] + o(/2),

(10.4.7)

where the constant c is the same as in (10.3.6). We stress that the probabilities in the brackets do not depend on the shifts of Qx and Q2 along the x axis. In the final result, (10.4.11), we will use the more emphatic notations tiOt (no hits on (O, t)n{d})

and fto, (no hits on (O, t) n {s}).

It remains to evaluate EPa5. Here most welcome is the observation that / 2 ), where

(10.4.8)

10.4 Averaging a combinatorial decomposition

265

Figure 10.4.3

Figure 10.4.4 Each of the segments joining an endpoint of v1 or v2 with an endpoint of / is of s type The explanation is that if a realization

say, then we have the situation shown infig.10.4.3. Because now vx is not hit by lines of {#J, the numbers of hits of the segments Qi&i and Q2^t coincide. Therefore on this set of realizations Pi\-\Q2>&t\) = 0(l2) as a result of subtraction of the distances. The same is true for •

,

and since P(C 1 ) = o(/), (10.4.8) follows. Fromfig.10.4.4 and (10.1.6) we also conclude that ) = 4^-/27To(0 + o(/ 2 ).

Lastly (compare with (10.4.7)) i = 2ctl2inQiQ4(A

n {s}) - UQiQ4(A n {<*})] 4- o(l2) (10.4.9)

Explanation We split B into subsets B = (Bn{s})v(Bn{d}). As can be seen fromfig.10.4.2(a), on B n {d} the segments Q20*j and Q 4 ^ are hit by lines from {gt} with probability 1 and do not contribute to a5, while the segments Q 3 ^ and Q^j are of d type. Together with (10.3.7) this yields

266

10 Sections through planar geometrical processes

= -2ctl2ftQiQ4(A

a5P(d

n {d}) + o(/2).

JSn{d}

A similar analysis is valid for B n {s}, hence (10.4.9). Since all other contributions to EPa5 are o(l2) we conclude that EPa5 = 4M2n0(t) - 2ctl2(tiQiQ4(A

n {d}) - UQiQ4(A n {s}))

2

+ o(l ).

(10.4.10)

Equations (10.4.4)-(10.4.7) and (10.4.10) together amount to a basic relation -po(f)=

-2Xno(t) + ctITOf(no hits on (0, 0 n {d}) - ctflot(no hits on (O, £) n {s}).

(10.4.11)

We stress that the events {d} and {5} can now be expressed in terms of the random angles (0l9 02) whose joint distribution is determined (marginally) by ftOf of (10.3.7). If 0X and 02 are defined as shown in fig. 10.5.2 then —, n

x —

(10.4.12)

w= 10.5 Further remarks on line processes I The result (10.4.11) can be generalized [44] to include probabilities of having k (rather than zero) intersections of the lines from {#J with a segment of length t. This generalization can be derived by the same method starting from a combinatorial decomposition similar to (10.4.3) but written for fi(Bk n C), where Bk = {g e G : k chords Xi hit g}. We leave it to the reader to check (using the algorithm (5.1.3)—(5.1.2)) that the desired decomposition can be obtained by making the following changes in the quantities a{ in (10.4.3): in al9 a2, replace Io by Ik; in a 3 , a5, replace Io by — 4_ x + Ik = AIk-u in a 4 , replace Io by / k _ 2 - 2Ik_x + Ik = A2/fc_2, where A and A2 are the first and the second differences, and 4 = 1 = 0

if the line segment in question is hit by exactly k chords %h otherwise (in each case the interpretation of a 'hit' is the same as in the description of/ 0 ).

10.5 Further remarks on line processes

267

The procedure of averaging repeats that of the previous section with minimal changes. The result is as follows: jt

2X{nk-t(t) - nk(t)) + ctA2UOt(k hits on (0,

t)n{d})

2

- ctA UOt{k hits on (O, t) n {5}),

(10.5.1)

where nk(t) = n

(there are k hits on (0, t))

(under both ft and n the interval (0, t) is taken on the x axis). II To obtain a check for (10.4.11) let us consider the case in which {#,} is randomly rotated T2-invariant Poisson. In the notation of §7.12, II (see also §10.1) we have Po(*) = -

^ Jo

exp(ex

P( -

cthot (no hits on (0, t) n {d}) - ctUOt (no hits on (0, t) n {s})

(10.5.2)

\2

Jo

Here /?(
and

- < 0x < n,

the condition is upon the angle of rotation. In our example the identically distributed angles 6X and 82 are conditionally independent and independent of the number of hits on the (0, t) interval of the x axis. On the other hand, as illustrated by fig. 10.5.1, we have up to o(&cp) X(cp + dcp) - k{(p) = ii{[AC]) - li{[AB~]) = li({CB\ n D4C]) - ii({CB\ n IAB\)

Hence,

A

B Figure 10.5.1 The length \AB\ = 1

268

10 Sections through planar geometrical processes

(10.5.3)

- p(cp))2 =

We multiply (2.11.2) by Q~k(
exp(-

dcp

= 2\ ap{-Hq>)t)f(. In view of (1O.5.2)-(1O.5.3) this coincides with (10.4.11). We could also obtain (10.5.1) if we used cxp(X(cp)(z - l)t) instead of

exp(-l(cp)t).

Ill As mentioned in §7.12 the marked intersection process {(xh i/^)} induced by {9i} on the x axis provides an adequate description of {#,}. In particular the sum of tlOt terms in (10.4.11) or (10.5.1) (and with it the probabilities nk(ij) can be expressed in terms of the distribution of {(xi9 ^ ) } . We now outline the corresponding derivation. Let us consider a pair of horizontal windows <5 1= =(-/,0)

and

S2 = (t,t + l)

situated on the x axis. The event F = {me JfG: there is exactly one pair of lines g1,g2em #! hits Sug2 hits S2}

such that

is a horizontal window counterpart of the event B in §10.3. We also consider two angles ij/l, \j/2 which are well defined whenever m e F (see fig. 10.5.2). The same argument which we used to demonstrate (10.3.6) applies in the proof that P(F) = cl2 + o(l2) with the same constant c as in (10.3.6). (The coincidence of the constants follows from the remark in §6.5 concerning the horizontal windows.) We define the horizontal window counterpart of the distribution tlOt as nS r (Ji, J2, A) = hm

P(r)

where Jt is a set in the space of values of \j/i9 [ J J is the corresponding event in JtG, i = 1, 2 and A cz JtG. 9\

r+ l Figure 10.5.2

10.6 Extension to random mosaics

269

The desired relation between tlot and ng f follows from the observations (a) we have obvious set relations in the space G {g : g hits vf under angle 0J = {g:g hits a horizontal window of length |cot ^ | • / under angle ^.} (b) we have an elementary identity |cot \//l cot \l/2\'(h — h) = c o t 'Ai c o t ^2Hence (see (10.4.12)) UOt(A n{d})-

UOt(A n{s}) = E* cot \j/1 cot i//2 IA(m)

(10.5.4)

where E* is the expectation corresponding to Fig,. This is a solution to our problem. We note that similar relations are valid also for the probabilities UOt which appear in the problems of §10.6 and §10.7. Among other things (10.5.4) implies the existence of E* cot ij/l cot ^ 2 . In the case in which the angles \jfx and i//2 are independent this expectation reduces to zero. (We have E* cot ij/i = E cot ^ = 0,

i = 1, 2

because each ^ has ^ sin \\i d\j/ density on (0,ri).see (7.15.7).)

10.6 Extension to random mosaics A remarkable feature of the calculations of §10.4 is that they remain valid for a broad class of random mosaics not necessarily generated by line processes. Apart from the M2-invariance property this class is determined by the further conditions (a), (b) and (c): (a) With probability 1 a mosaic should possess no vertices of 'multiple T type'(see fig. 10.6.1). (b) The process of edge midpoints of a mosaic should be of finite intensity and the distribution of the length of typical edge should possess finite mean. (c) The process of edges of a mosaic should not possess parallel pairs and should be of second order. Moreover, its second moment measure should possess a density in the sense of §7.14, Example 4. We shall denote the process of edges of the mosaic (a segment process) by {s,}.

7 T type

Double T type Figure 10.6.1 Multiplicities 1, 2, 3

Triple T type

270

10 Sections through planar geometrical processes

We briefly outline the necessary ^interpretations and changes from §10.4. Our first aim is to write the combinatorial decomposition for the set C nB, where C is as in (10.4.2) and B = {g e G: within R, g does not hit any edge s^. By virtue of (a) and the absence of edges on a line, this decomposition coincides with (10.4.3). Proof Let us denote by 0>t the points of intersection of edges s{ with dR, by {vi} the set of vertices of the mosaic which lie within R, and let Qi9 i = 1,..., 4 be as in fig. 10.4.1. Clearly, using the notation of §5.1, The four-indicator formula (5.1.2) yields zero whenever at least one of the points in a pair is of vt type (this could be not the case if we had vertices of multiple T type). Thus the decomposition will actually depend only on the pairs of points from {^} u {Qj, and the coefficients of the decomposition turn out to be the same as in (10.4.3). The explanations given there for the terms al9 a2, a4 and a5 carry forward if we now put Xt = R n st (compare with (10.4.1)). For a 3 , we obtain another expression: *3=-2

I

\Xt\,

(10.6.1)

5, hits both v 1 andv 2

which coincides with the old one if we consider the line process as a mosaic. Condition (b) allows us to conclude that EPa3 = - 2 V 2 E p ( M - t)+ + o(l2), (10.6.2) where Xo is {2n)~ times the intensity of edge centers, and p denotes the distribution of the typical edge length of the mosaic, l

x+ = x = 0

if x > 0 otherwise.

Proof We apply the construction of Example 4 in §9.2 to the edge process {st} = *n2{aS) and obtain its relative Palm distribution TImi. For it we write (9.1.4) with where s denotes that segment of the realization M~1{si} which lies on the x axis and is centered at O; |%| is the distance between the points Ms n vx and

Msnv2; A = {M eM2:Ms

hits both v1 and v2}.

The right-hand side of (9.1.4) is identically proportional to EPa3 while the left-hand side is asymptotically equivalent to the desired expression.

10.6 Extension to random mosaics

271

Expressions of the terms EP(a1 + a2) and EPaA retain the forms (10.4.5) and (10.4.7) with interpretations po(t) = P(no intersections on (O, t) with {sj), the segment (O, t) lies on the x axis, ftOr is the limiting (/ -• 0) distribution of {s^ conditional upon the event {there is one hit on each vh i = 1, 2 produced by different edges} whose asymptotical probability is c ^ 2 , see Remark to §10.3, IV. The events {d} and {s} are defined in terms of intersection angles 0 l5 02 as in (10.4.12). For EPa5 it is essential to check, that the considerations that led to (10.4.8) and are used in further analysis remain valid. For instance (10.4.8) now follows from the observation that if on fig. 10.4.3 vx is not hit by the edges of a mosaic then (If an edge of a mosaic hits a side of a triangle then at least one of the two other sides of the triangle is also hit by the edges of the mosaic). Therefore we again have EPa5 = 2(EPa3 — EPa4) + o(l2). In this way we come to a relation valid for random M2-invariant mosaics satisfying conditions (a), (b), (c): d -po(t)

- ZAOLV(\S\ - t)

+ c ^ f t ^ n o hits on (O,

t)n{d})

- cx tftOr(no hits on (O, t) n {5}).

(10.6.3)

By virtue of a general relation d — £ p (|s| - t)+ = p(|s| > t\ (10.6.4) (10.6.3) allows the calculation the distribution of the typical edge length s of the random mosaic in terms of the marked point process {(xh ^ ) } induced on the x axis (or any other test line). In particular, if in the {xh i/^J process we have

po(t) = exp(-ht),

h>0

and the angles ^ are independent, then the ft terms in (10.6.3) sum to zero (compare with (10.5.4)) and (10.6.4) yields From this we conclude that p (typical edge length exceeds t) = exp( — edge center intensity = nh2.

272

10 Sections through planar geometrical processes

A result equivalent to (10.6.3) has been obtained in a less rigorous setting in [3] based upon the Pleijel identity (6.9.5). Some corollaries for typical polygons of the mosaic have also been considered in [3].

10.7 Boolean models for disc processes The method of averaging the combinatorial decompositions seems to be far from exhausted on consideration of the preceding examples. A natural domain for expansion constitutes line and plane processes in IR3 but no work has been done in this area yet. However another possibility is to apply the technique to domain processes as suggested by the existence of combinatorial decompositions for members of Sylvester rings (see §5.4). Some work in this direction has already been started in chapter 10 of [3]. Below we consider essentially the same problem but we use a more rigorous approach in which we analyze the events that occur in a narrow rectangle R (see fig. 10.4.1). Let {bi} be an y 2 -invariant process of unit radius discs, and let P be its distribution. For simplicity we exclude multiple tangencies, i.e. we assume that with probability 1 in a realization no three discs have a common tangent line. We consider the Boolean model, i.e. the set (J b{. Boolean models for Poisson disc processes have been considered in §8.9. Now our {bt} is not assumed to be Poisson; however, we retain the white and black sets terminology, i.e. we call (J bt the black set and its complement white. The first step is to write the combinatorial decomposition for the set B nC, where C is as in (10.4.2) and B = {g e G:g nRis

completely white}.

Clearly B nC belongs to the Sylvester ring generated by the segments vx and v2 and the traces bt n R of the discs on R. The corresponding combinatorial decomposition (a version of (5.4.3)) will be of the form

where a1,..., a5 are versions of the terms in (10.4.3) while the terms a l9 a 2 and a 3 are principally new. In the present situation the expressions for al9 a2, a4 and a 5 given in §10.4 undergo only minor reinterpretations which reduce to a convention that now the points {^} are understood to be the intersection points of the circumferences of the discs with vx or v2 and Xt = bt n R.

The new expression for a 3 is as follows (compare with (5.4.3)): Ut(9))Ic(t(
(10.7.2)

273

10.7 Boolean models for disc processes

(b) Case s

(a) Case d

(c) Case d

bt

(d) Case s

*'k (e) Case d

(f) Case s

Figure 10.7.1 We refer to the configurations (a)-(b), (c)-(d) or (e)-(f) by mentioning the pairs J5-, J^; Qt, J^; or ^ J2), respectively

where t(cp) is the line tangent to dbt at the point with angular coordinate cp, and Iw = 1 if t(cp) n R is completely white, zero otherwise (w = dB if the discs are assumed open). We describe the terms a1? a 2 and a 3 using fig. 10.7.1. We have

«3 = Z 1 ^ ' ^ l U ^ i , Pj)Ic&t> m1* - h\ (10-7.3) Here Iw = 1 if the continuation within # of the segment in the argument is completely white, zero otherwise; Ic = 1 if the continuation of the same segment hits both (closures of) vx and v2. Under the assumption that the process of the disc centers is of second order and its second moment measure possesses a bounded density (in the sense of §7.14, Example 1), (10.7.1) can be averaged and the terms which are I2 by order can be separated. This procedure resembles similar operations in §10.4. The result depends on several Palm-type distributions of our disc process {bt} which can be obtained as limiting conditional distributions with conditioning events corresponding to the above diagrams. We denote by X < oo the intensity of black disc centers and start to consider separately the mean values of the summands in (10.7.1).

274

10 Sections through planar geometrical processes

-t

x

O

x + t

Figure 10.7.2

^

h Event {s} :

^ < | , *2 < |

Event {d):

^ < | , 02 >

|

Figure 10.73 If (O, t) is white then the events {s} and {d} are well defined The case of E(a1 + a2) can be treated as in §10.4. Then 2k Ea3 =

t

12

f°

II((x, x + t) is white) dx + o(l2)

J-t

where II is the relative Palm distribution of {bt} with respect to the process {Mj on the group M2 constructed in Example 3, §9.2. Thus with Improbability 1 we have a disc contacting the point O on the x axis, as shown in fig. 10.7.2. Explanation: assuming Qt = 0 we represent Ea3 as where g{M) is the line on which lies the directed segment representing M e M 2 , x(M) is the abscissa of the source of M, B(M) is the event 'the interval (— x(M), — x(M) + t) is white', E denotes expectation with respect to {Mi9 M^{bs}}. It remains to apply (9.1.4) and integrate using (3.13.3). Further Ea4 = t• Cl(t)- / 2 [fl Of ((O, t) is white f] {d}) - UOt((0, t) is white f] {5})] + o(l2) where asymptotically c^t) I2 is the probability of the event. There is exactly one pair of discs bi9 ty in realization such that db( hits vx

and

dty hits v2

and tlOt is the corresponding limiting conditional probability (see the Remark in §10.3, IV)). IIOt is a probability distribution on the space (0, 2n) x (0, 2n) x

Jiw,

with flOr-probability 1 the boundaries of two different discs pass through the points O and t on the x axis. The events {d} and {s} can be defined in terms of the angles i//1 and \j/2 as shown in fig. 10.7.3.

10.7 Boolean models for disc processes

O

275

*i

Event {d}

•+••*

Y

Event {d}

Event {s}

70.7.5 If the interval (O, x) is white then the events {d} and {s} are well defined

Also Ecu, = c2(t)'l2E^I(Oft)iswhiie(I{d}

- I[a])\Xl

- x2\ + o(l2)

where asymptotically c2(t) I2 is the probability of the event that in the expression of OL1 in (10.7.3) there is exactly one non-zero summand, E^^ stands for the expectation with respect to li^f, the corresponding limiting conditional distribution. This is a probability distribution in the union of two copies of the space (O, t) x (O, t) x J?R2; the copies correspond to the events {s} and {d}. With nj^-probability 1 we have two discs contacting the x axis at some points xl9 x2 e (0, t); see fig. 10.7.4. Lastly Ecc3 = c3(t)l2E^I(Oj)iswhitc(I{d}

- / { s } )x + o(/ 2 ),

2

where asymptotically c3(t)l is the probability of the event 'in the expressions of a 3 in (10.7.3) there is exactly one non-zero summand'. E^^ stands for the expectation with respect to Tl^f, the corresponding limiting conditional distribution. It is a probability distribution in the union of two copies of the space (0, In) x (O, t) x

Jtw.

With n^-probability 1 we have a disc bt with O e dbt and a disc bj contacting the x axis at a point x e (O, t) (see fig. 10.7.5). By symmetry we have EOL2 =

The case of

0.

10 Sections through planar geometrical processes

276

M

/w(

i 0.7.6

= E X 1(6. ^ is more complicated since the probability to have at least one pair Qi9 ^ for which Iw = 1 is O(/). Therefore it is necessary to perform subtraction of the terms corresponding to / s and Id. First we consider the event (a) in which the tangent line at ^ hits the opposite vertical window (fig. 10.7.6(a) shows one of the four components of the event (a)). The contribution of the event (a) in Ea5 is 2A/ 2 [n((O, t) is white) + I I ( ( - t , O) is white)] 4- o(/2). The contribution of the part of the complement of (a) which is defined by the relations ^ev2

and

lw(Qu 9S) = IW(Q2,,

^ev,

and

IW(Q3, Pt) = IW(Q4,,

= 1

or 2

(see fig. 10.7.6 (fe)) is o(/ ). The union of four components of the type (c) and of four components of the type (d) (see fig. 10.7.6) coincides with the event 'there is exactly one non-zero summand in the expression of a 3 in (10.7.3)'. The corresponding contribution to Ea5 equals -c3(t)l2tE^I(O,t)isy,hiu Putting all this together yields

(I{d} - I{s}) + o(l2)

10.8 Exponential distribution of typical white intervals

277

d — P(a segment of length t is white) at =—

n((x, x + t) is white) dx

- 2XU((-t9 0) is white) - 2>HI((O, t) is white) ((O, t) is white n {d}) - fto,((0, t) is white n {s})} r ) 1 , w h i t e M(/ w M

- / {l} M)(r - x).

(10.7.4)

10.8 Exponential distribution of typical white intervals If {bt} happens to be Poisson then each of the last three lines in (10.7.4) reduces to zero. We explain that annihilation of the two terms involving flOf follows from the remarks at the end of §8.9 and of §10.5, III. The E** vanishes by virtue of the following nature of the probability distribution Tl^f in the case of Poisson {bt}: the event {d} (or {s}), the variables xl9 x2 and m e J/R2 are independent; the projection of U%f on (O, t) x (O, t) is uniform while its projection on Jim is Poisson (identical with the distribution of The term E^^ vanishes for similar reasons. Correctly speaking, all terms in (10.7.4) depend solely on the distribution of the marked black interval process {Si9 \j/[91//"} (see the end of §8.9) induced by the Boolean model (J bt on a test line (say the x axis). Therefore it is natural to look for classes of processes {Si9 \l/'i9 ^-'} for which annihilations similar to the above occur. It was shown in [3] that if (a) the triads (|<5f|, ^-, \jt") for different values of i are independent (and certain moment exist) then the last three lines in (10.7.4) cancel. Now let us also consider the sequence {vj of white intervals (spaces between the black intervals 5t) induced by our Boolean model on the test line. We will assume in what remains that (b) the sequences {|vf|} and {(141, \jt'i9 ^")} are independent; (c) {|v£|} is a sequence of independent lengths. We stress that (a), (b) and (c) together yield a version of the renewal property as interpreted in §8.9. If (a), (b) and (c) hold then the distribution of the typical white interval length is necessarily exponential. Here is the derivation of this result whose prototype can be found in [3], chapter 10.

278

10 Sections through planar geometrical processes

We use the notation F(x) = P (typical |vj exceeds x). By virtue of the renewal properties of the process on the x axis n((O, t) is white) = U{(-t9 0) is white) = qF(t) and

ff

f°

Il((x, x + t) is white) dx = q\

F(t - x)F(x) dx,

where q = IT(the point O remains on the boundary of the black set). Then by a version of Palm formula (8.7.4) — P((0, t) is white) = at

-2kqF{t).

Therefore (10.7.4) becomes F(t) = t~1

F(t - x)F(x) dx.

(10.8.1)

Jo In terms of the Laplace transform

-JT

L(u)= I (10.8.1) is equivalent to

exp(-ut)F(t)dt

L'(U)=-L2(M),

i.e. L(u) = (u + C)" 1 or F(x) = exp( - Cx\ C > 0. This result is yet another illustration of the general principle according to which y 2 " i n v a r i a n c e often imposes rather heavy restrictions on the distributions of intersection processes (compare with §10.3, III).

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INDEX OF KEY WORDS

absolutely monotone function, 226 affine deformations group, 74, 75, 87, 88, 90 affine group A 2 , A 3 , 72, 77, 86, 89 affine group subgroup A°2, A j , 72, 77, 89 affine shape, 43, 82-6, 92-9, 241, 242 analysis of realizations, 137, 145, 161 anchored 1-lattice, 178 anchored polygon, 231, 233 anchored realization, 202 anchored renewal process, 211, 213 annulus, 139, 140 arc length measure, 43, 46, 48, 74 area measure, 51, 53 atom, 100, 101 averaging in the space of realizations, 192 ball process, 220 Boolean model, 214, 215, 252, 272 bounded lune, 107 breadth function, 15, 17, 18, 31, 33, 120, 121, 129 of a projection, 18 Buffon set, 103, 124, 255 Buffon's needle problem, 123 Buffon-Sylvester problem, 124 Cavalieri principle, 1, 3, 11, 40, 46, 51, 55-7 Cavalieri transformation, 2, 11, 21, 22, 24, 74, 214 chord, 1, 134, 141, 153, 156, 248 closed set, space of, 12, 13 cluster, 235 combinatorial decomposition, 101, 107, 262, 266, 270, 272 compactification, 71 companion system or set, 108, 111 composition of Lebesgue measures, 11, 22, 23, 24,27 consistent family, 168 covering lune, 105-7 Cox process, 173

Crofton's theorem, 18, 102 curvature radius, 31, 33 curve process, 199 Davidson hypothesis, 173, 179 Davidson inequality, 143 density function of a point process, 170, 171, 221, 222 differential notation of Haar measure, 8 of invariant measure, 8,173 directed flag, 44 disc process, 189, 272 displacement, small, 101, 112 distribution of a point process, 162, 204, 211, 217, 222, 225 distribution of typical mark, 181, 201, 204 distribution function of chord length, 156 of minimal angle, 158, 159 doubly stochastic Poisson process, 173, 184, 254, 255 drops, independent, 165, 174 elimination of a measure factor, 10, 47, 57, 80 elliptical space E, 12, 21, 23, 24, 96, 99, 104 equation with Palm distribution, 217, 220, 256 Euclidean shape, 43 Euler formula, 121 factor measures, 28, 35, 37, 57, 59, 61 Fatou lemma, 197, 261 fibered space, 10, 21, 23, 45 flag, 117 directed, 44, 45 representation, 100, 118-22, 126 formula, four indicator, 101, 112, 270 Gauss-Bonnet theorem, 103, 107, 126 generator, 3, 10, 21, 27

284

Index of key words

geodesic lines on a sphere, 46, 50, 104 group AS, 72, 77 A?, 83 Ci,72-5 C(1), 86-8 C(2), 87, 88 C 2 ,88 H x ,72-5 H 2 ,87 Mn of Euclidean motions of Rn, 34, 47 ¥„ of translations of R", 3, 9 W 2 of rotations of R2, 43, 74 W 3 of rotations of R3, 44 of affine deformations, 74, 75, 87, 88, 90 of affine transformations, 72, 77, 86, 89 of 'independent shifts', 38, 41 Haar factorization, 5, 63, 144, 191 Haar measure, 5, 9, 44, 55, 61, 89, 137 bi-invariant, 6, 9, 46, 74, 76, 77, 87, 89, 96 finite, 9 left-invariant, 5, 9, 75 right-invariant, 5, 75, 89 on A 2 , 77 on A 3 , 89 on A2, 76 on A?, 89 on Mn, 47, 61, 66, 78, 89 on W 3 , 45, 59 homothety, 9,10 horizontal window, 268

Laplace operator, 33 Laplace transformation, 226, 278 lattice randomly rotated, 177, 207 randomly shaped, 177,180 randomly shifted, 175, 176 truncated random, 180 Lebesgue dominated convergence theorem, 195, 260 Lebesgue factorization, 3, 35, 40, 200, 204, 234 length intensity, 188, 195, 196 line at infinity, 99 line process, 189, 191, 229, 252, 255, 256, 258, 262, 266 locally-finite measure, 4 logarithmic measure, 9, 74 lune, 104, 105, 107

marked point process, 181, 187, 200, 203, 228, 232, 234, 243 angular marks, 252, 258, 268 marked segment process, angular marks, 215, 216, 252, 277 measure-representing model, 15, 24, 25, 26, 27, 29, 35, 45 measures generate pseudometrics, 17 Mecke's proof, 143 metric, 19, 32, 33 Meusnier theorem, 122 minimal angle, distribution function, 158 Minkowski proposition, 16, 17, 30, 33 Mobius band, 13, 21 independent angles, 183, 186, 197, 198, 216, moment measure, 187 233, 258, 259, 267, 269, 271, 277 first, 38, 187, 201, 204, 206, 234, 239-42, independent drops, 69 245, 246 independent marks, 181, 202 second, 38,188,190,197, 198, 206, 259, infinitesimal domain, 138,139, 194, 201, 205, 269, 273 214, 234, 260, 264 mosaic, 150 intensity (finite), 187, 193, 200, 212, 213, 221, multiple T-type node, 269 227, 232, 235, 240, 243, 252, 253, 255, multiprojection, 128, 129 269, 271 intersection (trace) process, 190, 196-8, 215, needles, d-type, s-type, 108, 115, 124, 262, 263 216, 252, 266, 268, 271, 277, 278 node, 146, 147, 149, 150, 186, 197, 235 invariant measure, 3, 234 of order two, 147, 150, 151, 186 on E and E, 53, 54 non-zonoidal convex body, 33 on G and G, 47, 49, 50, 59, 101, 124, 234, 252, 263 ordering map, 164 on T and T, 51, 52 outer lune, 105, 113 the, 18, 56 invariant point process, 175, 227, 235, 240 pairs of lines isoperimetric inequality, 153, 155 in R3, 61 on R2, 60 Jacobian calculation, 7, 8, 48, 51, 58, 76, 81, pairs of planes in R3, 60 84,91 pairs of points in IR2, 58 in R3, 59 kinematic measure, 55, 59 on a sphere, 59 /c-subsets, 163 Palm distribution, 200, 204, 205, 209, 211, 212, 220, 223, 228, 252, 257 labeled triangle, 62, 70

Index of key words of a line process, 253, 254, 256, 257 relative, 213, 214, 227, 228, 232, 236, 237, 239, 253, 270, 274 Palm formulae, 207, 208, 210, 216-18, 223, 225, 255, 277 Palm-mixing property, 258, 259 Palm-type distribution, 275-7 horizontal windows, 268, 274 vertical windows, 259 parent map, 235-8 PIA sin property, 186, 197 Pleijel identity, 153, 272 Poincare-Blaschke formula, 145 point process on a group, 227-30, 235, 238, 240, 243, 253 point processes, jointly distributed, 227 Poisson cluster process, 184, 185, 208, 217 Poisson line process (mosaic), 175, 183, 186, 191, 231, 233, 235, 238-40, 244, 247, 255, 256 Poisson point process, 167, 168, 170, 173, 175, 182, 183, 199, 206, 207, 217, 229, 230, 239, 241-3 doubly stochastic, 173,184, 254, 255 in a strip, 182 Poisson process of balls (discs), 184,189, 215, 216 Poisson segment process, 185, 213, 215 polar map, 46 polygon process, 245 probability continuation, 162, 164, 166 process of non-intersection and non-interaction balls, 220 product model, 24, 26, 27, 29, 48 product representation of invariant measure, 29, 172 projection, 18, 30, 33, 34,104,113,130, 233 of a measure, 4, 236, 242, 246 of a point process, 202, 240-2 pseudometric, 16, 17 quadruples of planes in R3, 66 quadruples of points in IR2, 82, 84 in IR3, 66 quintuples of points in IR3, 91, 92 Radon ring, 100, 111, 125 random line process, 173, 180, 183, 229, 256, 266 random measure, 37, 38, 42, 247 random mosaic, 185, 186, 191, 192, 197, 231, 235, 244, 245, 252, 269, 271 random polygons, 247, 249 randomly rotated line process, 254, 267 realization, 5, 138-40, 145-52, 161, 162, 228 relative density function, 169,170 relative Palm distribution, 213, 214, 227, 228, 232, 237, 253, 270, 274 renewal process, 211, 226

285

renewal segment process, 213-15, 226, 277 rose of directions, 25, 116, 123, 142, 183 rose of hits, 30, 33, 142 Schwartz inequality, 39 segment process, 173, 185, 186, 190, 198, 199, 213, 214, 230, 261, 269 semiring, 163, 171, 172 shape, 60, 61, 66, 67, 69, 90, 243 shape density for tetrahedrons, 67, 90 for triangles, 63-5, 68, 82, 158 Shepp example, 173 shield, 39, 40, 172, 255 Siegel theorem, 179 signed measure, 130 simplex, 163, 243 simplexial shape, 69, 94, 243 size parameter, 59, 61, 63, 66, 244 skeleton, 108, 109 space of labeled affine shapes, 85, 92-9 space of labeled triangular shapes, 63, 69, 70 stereographical projection, 71 stochastic construction, 227, 233, 236-8, 245, 247 strings,rf-type,s-type, 109,110, 273 support line, 15 Sylvester problem, modified, 72, 84-6, 94, 109 Sylvester fing, 109,272 symmetry principle, 48, 51-3 tangent bundle, 11, 23 tetrahedral shape, 66, 67, 69, 90 thinning, 180, 202, 238, 240-3 independent, 220 three points on a sphere, 68 total measure, 68-70, 85, 93, 95,132,159, 160, 178 translational counterpart of integral of means curvature, 122 triads of lines in R2, 62, 238 triads of planes in IR3, 61 triads of points in IR2, 63, 78, 246 in IR3, 65 triangular shape space, 63-71 typical edge length distribution, 270, 271 typical mark, 181, 200, 201, 204, 231, 232, 235 typical w-tuple, affine shape distribution of, 242 typical polygon, equally weighted, 231, 233, 245, 250 typical polygon, ^-weighted, 235-8, 250 typical simplex, shape distribution of, 243 typical triangle, perimeter-shape distribution of, 240-2, 244 typical vertex, shape of, 245-7 typical white interval, Boolean model, 276 uniqueness class, 168, 169, 173

286

Index of key words

variational distance, 219 vertical window, 133, 134, 152, 259, 262 Voronoi mosaic, 244, 250 weak convergence, 119

wedge, wedge function, 111, 113-15, 117, 118 zonoid, 17, 116, 117, 119-21, 129, 131 zonoid equation, 130 zonotop, 119, 120