Factorization Calculus and Geometric Probability

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

EDITED BY G.-C. ROTA

Volume 33

Factorization calculus and geometric probability

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Luis A. Santalo Integral Geometry and Geometric Probability George E. Andrews The Theory of Partitions Robert J. McEliece The Theory of Information and Coding: A Mathematical Framework for Communication Willard Miller, Jr Symmetry and Separation of Variables David Ruelle Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics Henry Mine Permanents Fred S. Roberts Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences L.C. Biedenharn and J.D. Louck Angular Momentum in Quantum Physics: Theory and Application Lawrence C. Biedenharn and James D. Louck The Racah-Wigner Algebra in Quantum Physics: Theory and Application John D. Dollard and Charles N. Friedman Product Integration with Applications to Differential Equations William B. Jones and WJ. Thron Continued Fractions: Analytic Theory and Applications Nathaniel F.G. Martin and James W. England Mathematical Theory of Entropy George A. Baker, Jr and Peter R. Graves-Morris Pade Approximants, Part II: Extensions and Applications E.C. Beltrametti and G. Cassinelli The Logic of Quantum Mechanics G.D. James and A. Kerber The Representation Theory of the Symmetric Group M. Lothaire Combinatorics on Words H.O. Fattorini The Cauchy Problem G.G. Lorentz, K. Jetter and S.D. Riemenschneider Birkhoff Interpolation Rudolf Lidl and Harald Niedereitter Finite Fields W. T. Tutte Graph Theory Julio R. Bastida Field Extensions and Galois Theory John Rozier Cannon The One-Dimensional Heat Equation Stan Wagon The Banach-Tarski Paradox Arto Salomaa Computation and Automata N. White (ed) Theory ofMatroids N.H. Bingham, CM. Goldie and J.L. Teugels Regular Variation P.P. Petrushev and V.A. Popov Rational Approximation of Real Functions N. White (ed) Combinatorial Geometries M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory J. Aczel and J. Dhombres Functional Equations in Several Variables M. Kuczma, B. Choczewski and R. Ger Iterative Functional Equations R. V. Ambartzumian Factorization Calculus and Geometric Probability G. Gripenberg, S.-O. Londen and O. Staffans Volterra Integral and Functional Equations George Gasper and Mizan Rahman Basic Hypergeometric Series

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Factorization calculus and geometric probability R. V. AMBARTZUMIAN Institute of Mathematics, Armenian Academy of Sciences

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CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521345354 © Cambridge University Press 1990 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1990 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Ambartzumian, R. V. Factorization calculus and geometric probability/R. V. Ambartzumian. p. cm.-(Encyclopedia of mathematics and its applications; v. 33) ISBN 0 521 34535 9 1. Stochastic geometry. 2. Geometric probabilities. 3. Factorization (Mathematics) I. Title. II. Series. QA273.5.A45 1990 519.2-dc20 89-7314 CIP ISBN 978-0-521-34535-4 hardback ISBN 978-0-521-08978-4 paperback

CONTENTS

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18

Preface

ix

Cavalieri principle and other prerequisities The Cavalieri principle Lebesgue factorization Haar factorization Further remarks on measures Some topological remarks Parametrization maps Metrics and convexity Versions of Crofton's theorem

1 1 3 5 9 10 14 15 18

Measures invariant with respect to translations The space G of directed lines on R2 The space G of (non-directed) lines in U2 The space I of oriented planes in U3 The space E of planes in U3 The space T of directed lines in U3 The space r of (non-directed) lines in IR3 Measure-representing product models Factorization of measures on spaces with slits Dispensing with slits Roses of directions and roses of hits Density and curvature The roses of T3-invariant measures on E Spaces of segments and flats Product spaces with slits Almost sure T-invariance of random measures Random measures on G Random measures on E Random measures on T

20 20 21 22 23 23 24 24 27 28 29 30 31 34 36 37 38 40 41

vi

Contents

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18

Measures invariant with respect to Euclidean motions The group W 2 of rotations of IR2 Rotations of U3 The Haar measure on W 3 Geodesic lines on a sphere Bi-in variance of Haar measures on Euclidean groups The invariant measure on G and G The form of dg in two other parametrizations of lines Other parametrizations of geodesic lines on a sphere The invariant measure on r and F Other parametrizations of lines in IR3 The invariant measure in the spaces E and E Other parametrizations of planes in U3 The kinematic measure Position-size factorizations Position-shape factorizations Position-size-shape factorizations On measures in shape spaces The spherical topology of £

43 43 44 45 46 47 47 48 50 51 52 53 53 55 57 59 61 67 70

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16

Haar measures on groups of affine transformations The group A5 and its subgroups Affine deformations of IR2 The Haar measure on A^ The Haar measure on A 2 Triads of points in IR2 Another representation of d(r)V Quadruples of points in U2 The modified Sylvester problem: four points in IR2 The group A3 and its subgroups The group of affine deformations of IR3 Haar measures on A 5 and A 3 V3-invariant measure in the space of tetrahedral shapes Quintuples of points in IR3 Affine shapes of quintuples in 1R3 A general theorem The elliptical plane as a space of affine shapes

72 72 74 76 77 78 80 82 84 86 87 89 90 91 92 94 96

5 Combinatorial integral geometry 5.1 Radon rings in G and G 5.2 Extension of Crofton's theorem 5.3 Model approach and the Gauss-Bonnet theorem 5.4 Two examples 5.5 Rings in E 5.6 Planes cutting a convex polyhedron 5.7 Reconstruction of the measure from a wedge function 5.8 The wedge function in the shift-invariant case

100 100 102 103 108 111 113 114 115

Contents

5.9 5.10 5.11 5.12

Flag representations of convex bodies Flag representations and zonoids Planes hitting a smooth convex body in U3 Other ramifications and historical remarks

vii

118 119 120 123

6 Basic integrals 6.1 Integrating the number of intersections 6.2 The zonoid equation 6.3 Integrating the Lebesgue measure of the intersection set 6.4 Vertical windows and shift-invariance 6.5 Vertical windows and a pair of non-parallel lines 6.6 Translational analysis of realizations 6.7 Integrals over product spaces 6.8 Kinematic analysis of realizations 6.9 Pleijel identity 6.10 Chords through convex polygons 6.11 Integral functions for measures in the space of triangular shapes

127 127 130 131 133 134 137 141 145 153 156 158

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15

161 161 162 165 168 171 173 174 175 177 177 179 181 185 187 192

Stochastic point processes Point processes /c-subsets of a linear interval Finite sets on [0, b) Consistent families Situation in other spaces The example of L. Shepp Invariant models Random shift of a lattice Random motions of a lattice Lattices of random shape and position Kallenberg-Mecke-Kingman line processes Marked point processes: independent marks Segment processes and random mosaics Moment measures Averaging in the space of realizations

8 Palm distributions of point processes in U" 8.1 Typical mark distribution 8.2 Reduction to calculation of intensities 8.3 The space of anchored realizations 8.4 Palm distribution 8.5 A continuity assumption 8.6 Some examples 8.7 Palm formulae in one dimension 8.8 Several intervals 8.9 T x -invariant renewal processes 8.10 Palm formulae for balls in Un 8.11 The equation II = 0 * P

200 200 202 202 204 205 207 208 210 211 216 217

viii

Contents

8.12 8.13 8.14

Asymptotic Poisson distribution Equations with Palm distribution Solution by means of density functions

218 220 221

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

Poisson-generated geometrical processes Relative Palm distribution Extracting point processes on groups Equally weighted typical polygon in a Poisson line mosaic Solution Derivation of the basic relation Further weightings Cases of infinite intensity Thinnings yield probability distributions Simplices in the Poisson point processes in Un Voronoi mosaics Mean values for random polygons

227 227 229 231 232 234 235 238 240 243 244 247

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Sections through planar geometrical processes Palm distribution of line processes on R2 Palm formulae for line processes Second order line processes Averaging a combinatorial decomposition Further remarks on line processes Extension to random mosaics Boolean models for disc processes Exponential distribution of typical white intervals

252 252 255 256 262 266 269 272 276

References Index

279 283

PREFACE

The subject of this book is closely related to and expands classical integral geometry. In its most advanced areas it merges with those topics in geometrical probability which are now known as stochastic geometry. By the application of a number of powerful yet simple new ideas, the book makes a sophisticated field accessible to readers with just a modest mathematical background. Traditionally, integral geometry considers only finite sets of geometrical elements (lines, planes etc.) and measures in the spaces of such sets. In the spirit of the Erlangen program, these measures should be invariant with respect to an appropriate group acting in basic space - to ensure that we are still in the domain of geometry. Assume that the basic space is Un (as is the case in the most of this book). If the group contains translations of Un, then the measures in question are necessarily totally infinite and cannot be normalized to become probability measures. Yet a step towards countably infinite sets of geometrical elements changes the situation: spaces of such sets admit probability measures which are invariant and these measures are numerous. The step from finite sets to countably infinite sets directly transfers an integral geometrician into the domain of probability. The vast field of inquiry that opens up surely deserves attention by virtue of the mathematical elegance of its problems and as a potentially rich source of models for applied sciences. Probability theory offers a ready apparatus for the study of countably infinite random sets, i.e. random point processes. In stochastic geometry their realizations should lie on manifolds which represent the spaces of the geometrical objects in question. The book begins with chapters on integral geometry. The preparation of the aforementioned step to geometrical point processes dictated the choice of the content of these early chapters, with the result that they have very little overlap with earlier books on integral geometry, including the fundamental

x

Preface

book by Santalo [2]. Chapter 1-4 can be considered as an introduction to Haar factorization, a rather neglected subject within integral geometry. Chapter 5 is devoted to combinatorial integral geometry (a complete account of this novel theory can be found in the author's earlier book [3]). A variety of geometrical integrals is presented in chapter 6. Our preoccupation with factorization of measures is justified on further development: a version of this tool plays a major role in the study of geometrical random processes with laws which are invariant under the action of the prescribed group. This is the way we approach the fundamental notion of Palm distribution which is a key to the solution of many problems given here. The content of the chapters on geometrical processes ranges from presentation of basic notions and examples in chapter 7, via Palm distribution theory for point processes in Euclidean spaces (chapter 8) and Palm distributions for point processes on groups (chapter 9), to the synthesis of many previous ideas in chapter 10. Among the innovations which contributed to a simplified treatment of the whole subject we mention the following: (1) The systematic use of the so-called Cavalieri principle which replaces many Jacobian calculations by clear geometrical reasoning. (2) Strong reliance on the theorems of uniqueness of Haar measures on groups. In particular this enables us to give simple rigorous proofs of uniqueness of the standard invariant measures in the spaces of lines, planes etc. (3) Derivation of Haar measures on groups based on knowledge of Haar measures on their subgroups. (4) Integral geometrical analysis of'realizations' with later use of the results in stochastic contexts. (5) A new treatment of the problem of probabilistic description of 'typical elements' generated by geometrical processes by reduction to calculation of the intensities of partial (thinned) point processes. (6) Rigorous application of the results of combinatorial integral geometry to problems involving geometrical processes. (The first attempt in this direction was made in [3].) Apart from making things simpler these new ideas yield new approaches to a number of long-standing problems: random shapes, J. J. Sylvester's Vierpunktproblem, translation-invariance with probability 1, zonoids and others. Doubtless zonoids belong to the basic notions of translatory integral geometry and this theme occurs in many sections of the book. Readers will surely notice that we often treat the planar and spatial cases separately, or even restrict ourselves to just the planar case, even if a single

Preface

xi

treatment in general Un may seem possible. In some cases this is because of the author's desire to make things clearer but in others the spatial case simply remains unexplored. Thus with some justification the book can claim it is a research monograph in a new field. However, as regards level and style it resembles a textbook. Perhaps this duality complies with the purposes proclaimed by the Editor for this Encyclopedia. The final stages of preparation of the book took place during the blockade of Armenia which was the result of the struggle to end the separation of Karabach from Armenia. The problem of conveying material between the publisher and the author could not have been solved under the circumstances without the vital shuttle missions of Professor C. Mutafian from Paris. It is the author's duty to acknowledge both this and the absolutely essential help offered with unfading enthusiasm by V. K. Oganian at all phases of the work.

Cavalieri principle and other prerequisites

The aim of this chapter is to present some basic mathematical tools on which many constructions in the subsequent chapters depend. Thus we will often refer to what we call the 'Cavalieri principle'. We try to revive this old familiar name because of the surprising frequency with which the transformations Cavalieri considered about 350 years ago occur in integral geometry. No less useful will be the principles which we call 'Lebesgue factorization' and 'Haar factorization'. The first is a rather simple corollary of a well-known fact that in Un there is only one (up to a constant factor) locally-finite measure which is invariant with respect to shifts of Un, namely the Lebesgue measure. Haar factorization is a similar corollary of a much more general theorem of uniqueness of Haar measures on topological groups. We use the two devices in the construction of Haar measures on groups starting from Haar measures on subgroups. Integral geometry binds together such notions as metrics, convexity and measures, and these interconnections remain significant throughout the book; §§1.7 and 1.8 are introductory to this topic.

1.1 The Cavalieri principle The classical Cavalieri principle in two dimensions can be formulated as follows. Let D1 and D2 be two domains in a plane (see fig. 1.1.1). If for each value of y the length of the chords Xx and X 2 coincide, then the areas of Dx and D2 are equal. The proof of this beautiful geometrical proposition follows from the representation of the area of Dh i = 1, 2, by the integral

1 Cavalieri principle and other prerequisites

-Xi-

Figure 1.1.1 Xt is the intersection of Dt with the horizontal line on the level y

X f dy. \

Pairs of domains having the above property arise whenever we consider transformations of the plane of the type x1 = x + a(y) yi = y>

which are clearly area-preserving. To these transformations the following interpretation can be given. We consider the plane as composed of 'rigid' horizontal lines. The transformation rigidly shifts each horizontal line along itself (although the shifts can be different for different lines). The domain D2 in the above example can be considered to be the image of Dx under some transformation of this type. We call this a Cavalieri transformation. Measures in U2 which remain invariant with respect to Cavalieri transformations are numerous. For instance, measures given by the densities of the form f(y)dxdy are all invariant as shown by the identity f(y) dx dy =

fWX^y)

dy=\\

f(y) dx dy,

(1.1.1)

where Dx and D2 are as in fig. 1.1.1. Similarly (Fubini theorem) it can be shown that any product measure m x Ll9 where L x is the Lebesgue measure on the Ox axis and m is any measure on the Oy axis, is invariant with respect to Cavalieri transformations of U2. In the sequel similar transformations of other product spaces will occur. The typical situation here will be as follows.

1.2 Lebesgue factorization

3

Let the basic space X be a product of two spaces X = YxH*, (xsX,yeY,0>eUk) (the second factor is /c-dimensional Euclidean). Assume that a transformation of the space X has two properties: (a) it sends each generator, i.e. the set y x Uk = {(y, 0>):yis constant, 9 changes in Uk}, into the same generator; (b) it preserves the distances between the points of a generator (i.e. the generators are 'rigid'). Such transformations we again call Cavalieri. The Cavalieri principle in this situation is as follows. Every product measure m x Lfc, where Lk is the Lebesgue measure on Uk and m is any measure on Y, is invariant with respect to Cavalieri transformations of X. In the spaces of importance described in chapter 2 we actually have natural groups of Cavalieri transformations.

1.2 Lebesgue factorization We consider a product of two spaces X = Y x U\ k where U is /c-dimensional Euclidean space, while the space Y here remains unspecified (any separable metric space will suffice). Let Tfc be the group of translations of Uk. We define the action of a translation t e Jk on the space X as follows. For (y9 9\ where yeY,0>eMk, we put t(y, &) = (y, t») (this means that t is y-preserving). A measure /i on X is called invariant with respect to Jk (or simply Tfcinvariant) if for every t eJk and C e X we have li(tC) = n(C)9 where tC = {t(y9 9): (y, 9) e C}. (1.2.1) To check (1.2.1) it is enough to consider the product sets, i.e. to take C = Ax B, i c Y , B c R\ in which case (1.2.1) reduces to li(t(A x B)) = fi(A x tB) = fi(A x B\ where tB denotes the translation of B by t: tB = {t0>:0>e B}.

4

1 Cavalieri principle and other prerequisites

As usual, measures which are finite on compact sets we call locally-finite. We say that a measure monX has a locally-finite projection on Mk, if for every compact B a Uk we have m(Y x B) < oo. The Lebesgue factorization principle states that: Any locally-finite and Tfc-invariant measure \i on X = Y x IRfc is necessarily a product measure: fi = m x Lfc, where Lk is Lebesgue measure on Uk and m is a locally-finite measure on Y. If additionally m has a locally-finite projection on IRfc, then H = XPx

Lfc,

where X ^ 0 is a constant and P is a probability measure on Y, i.e. P(Y) = 1. Proof Let us fix a set A o c Y which has compact closure and let us regard H(A0 x B) as a set function depending on #. It follows from the properties of fi that this is a measure on Y which is translation-invariant and locally-finite. It is known from analysis that any such measure is proportional to Lebesgue measure, i.e. fi(A0 xB) =

m(A0)'Lk(B).

So far m(A0) has been some constant which does not depend on B, but may depend on our choice of Ao. Now we fix a set Bo c IRk which has compact closure and consider H(A xB0) =

m(A)'Lk(B0)

as a function of A. Clearly fi(A x J50) is a measure on Y. This implies that m is a locally-finite measure on Y. This proves the first assertion. In the case where \i has a locally-finite projection on Uk, we have i.e. m(Y) < oo. We get the second assertion when we put (assuming k > 0) A = m(Y), P = r1m. In the factorization table 2.8.1 we give several important examples where Lebesgue factorization is directly applied. We mention, however, that the factorizations of table 2.9.1 are valid under quite different conditions: in the corresponding spaces the group of shifts no longer transforms product sets into product sets. Remark on terminology In this book we consider only locally-finite measures. However, in the text we often omit the adjective 'locally-finite'. Thus a 'measure' will always mean a 'locally-finite measure'.

1.3 Haar factorization

5

1.3 Haar factorization The main idea of Lebesgue factorization can be extended to product spaces where one of the space factors is a group. For a broad class of locally-compact topological groups (an exact account of the theory can be found in [4]) an important theorem is valid which establishes existence and uniqueness (up to a constant factor) of the so-called left-invariant and right-invariant Haar measures. In general the two measures need not be proportional. When they are, we have the bi-invariant Haar measure (which is again defined up to a constant factor). We will always tacitly assume that our groups belong to the class mentioned above, as do all the concrete groups we consider in this book. Let U be a group. A non-zero measure /i on QJ is called left-invariant Haar if h(uA) = h(A)

(1.3.1)

for a r b i t r a r y w e U a n d A c (U. H e r e uA = {uux:

MX

e A},

where uu1 denotes group multiplication. A non-zero measure h is called right-invariant Haar if h(Au) = h(A)

(1.3.2)

for any A a U and ueV. Here Au = {uxu : u1 e A). (

We will use the notation /z y, /ijj and hv, respectively, for left-, right- and biinvariant measures on U. By essentially repeating the proof of the previous section we can extend its result to product spaces X = Y x U, where the factor U is a group (it replaces Uk), Y again is a separable metric space. Any measure \i on X which is invariant with respect to the transformations

u1(y,u) = (y,ulu) necessarily factorizes:

H=m x

fcff,

(1.3.3)

where m is some measure on Y. If we use right-multiplication, i.e.

u1(y,u) = (y,uu1), then the right-invariant Haar measure h({j will appear in (1.3.3). Below we will refer to these factorizations as 'Haar factorizations' Remark In chapters 8 and 9 (in the point processes context) we apply the above proposition in the situation in which Y is the space of 'realizations'.

6

1 Cavalieri principle and other prerequisites

There we gloss over the question of introducing the metric on such Y. (This work has been carried out in detail in [18].) In some cases we can apply Haar factorization to find Haar measures explicitly, as well as to obtain the criteria of existence of bi-invariant Haar measures (i.e. measures which satisfy both (1.3.1) and (1.3.2)). Let X be a (non-commutative) group, and let V and V be two subgroups of X. Assume that each xeX admits both representations x = UiVr,

M, e U ,

x = vxuT,

uT e U ,

1

'

vreV

'

(1.3.4)

vxeV

and that each of these representations is unique. (The letters T and V in the subscripts stand for 'left' and 'right'.) According to (1.3.4) the set-theoretical product U x V can be mapped on X in two ways: / i1 :

(M, V)

f2 : (M,

- • uv

V) -•

(1.3.5)

vu

and these maps are one-to-one. In other words, the product U x V can be used as a model for X in two different ways:

(«.,fr) = /r 1 w (ut,vl)=f2-1(x), where f'1 Now

denotes the inverse of/.

(ur,vvl% veV. Therefore the left-invariant Haar measure h$ necessarily admits two Haar factorizations, namely fig {k fcg x mx

*2 & m2 x

fc«

(L16)

where m1 and m2 are some measures on V and U, respectively. The symbol = denotes the image of the measure under the map /. There are similar equations for the right-invariant Haar measure on X:

fc« 4 m\ f,(r) il

L(r)

x

h§ /

yi.J-i)

where m[ and m'2 are some measures on U and V, respectively. In the cases where the Haar measures on the subgroups U and V are known, the partial information given by these equations can be used for the purpose of finding Haar measures on X. Some examples are given in chapter 4. The maps fx and f2 can be used to formulate a necessary and sufficient condition of bi-invariance of hx- By repeated application of Haar factorization

1.3 Haar factorization

7

we conclude that any measure on X which is invariant with respect to the transformation x —• uxv

is necessarily the image under fx of the measure where c is a constant. Similarly, any measure on X which is invariant with respect to the transformations is necessarily the image of the measure

chyxhfl (perhaps with a different constant) under f2. These images can be substantially different. But let us assume that both images are proportional to a measure h

on X. For any i c X w e will have h(u2vlAu1v2)

= h(vlAul) = h(A).

Since both u2v1 and u1v2 represent general elements from X, this is essentially the condition defining the bi-invariant Haar measure on X. We have come to the following result. On U x V we consider two measures:

h$ x fc« and hfi x fcg>. Their respective images under fx and f2 are proportional if and only if there exists a (unique) bi-invariant measure hx on X, hx being proportional to the above-mentioned image measures. The practical application of this criterion can be as follows. Each of the pairs (ul9 vT) or (uT, vx) can serve as coordinates on the group X. We express ux and vr in terms of uT and vx (both pairs of variables correspond to the same x as in (1.3.4)): 1

^lV^

X)

(1.3.8)

vr = (p2(uT9 VJ.

We can assume that fx is trivial, i.e. (ul9 vT) = x. l

Then f2 is given by (1.3.8). Now application of the above criterion reduces to the usual Jacobian calculation, i.e. to a check that the transformation (1.3.8) maps hfjj x h§ into c • h$ x hy\ In all the cases we consider in this book, bi-invariance of hx implies that the constant c equals one. The typical situation will be as follows. The elements w e U and veV will depend on a finite number of parameters. Therefore c will equal the absolute value of the determinant of a matrix which

1 Cavalieri principle and other prerequisites

we briefly denote by du

dv

du

dv

c =

Since c is a constant it is enough to calculate the value of this Jacobian at the points u = lu

(the unit element of QJ)

v = lv

(the unit element of V).

and We have d(px(u, v)

dcp^u.v)

du

dv

dcp2(u, v)

dcp2(u, v)

du

dv

dv

du dq>2(u9

9

du

v)

dv

Because QJ and V will always be groups of transformations of the same space, from (1.3.4) we find (piiu, l y ) = M, r = d(r)wr d(\.

(1.3.9)

This corresponds to writing a measure in terms of coordinates.

Table 1.3.1 Group

Element xe

Bi-invariant Haar Left-invariant Haar Right-invariant Haar d(rb
S ! X (0, ft),

as shown in fig. 1.6.1. The image (v, Q>) is defined for all points at e S 2 except for the 'poles' N and S. In a typical situation a parametrization map of a space X onto a space Y will be a homeomorphism between their slitted versions X\S!->Y\S2,

(1.6.1)

where the excluded sets S1 and S2 will be less than X or Y in dimensionality. As soon as such a map is specified we will write X « Y.

(1.6.2)

Let m be a measure on X. The image of m under parametrization (1.6.1) will provide an adequate description of m whenever m(S1) = 0.

(1.6.3)

Yet in general not every measure m1 on Y for which m1(S2) = 0

(1.6.4)

can be considered to be an image of a measure on X (recall that in our usage measures are necessarily locally-finite). Clearly the map converse to (1.6.1) can send a non-compact set B c Y into a subset of a compact set in X. Therefore a measure mx on Y happens to be

l-COS V

The image of CJ

Figure 1.6.1 The circumcylinder is S x x (— 1, 1)

1.7 Metrics and convexity

15

an image of a measure on X whenever an additional condition m^B) < oo

(1.6.5)

is satisfied for every B c Y \ S 2 with the described property. If both (1.6.4) and (1.6.5) are satisfied, then m1 in a sense represents a measure on X. For instance, the measure on S x x (0, n) given by the density (v)"1 dv dO is locally but not totally finite and fails to represent a measure on § 2 - The measure sin v dv dO represents the area measure on S 2 . The precautions (1.6.3)—(1.6.5) would be pointless if we could complement the map (1.6.1) by a one-to-one map between S1 and S2 with the property that the one-to-one map between X and Y that arises sends a compact C cz X into a relatively compact set C ' c Y and vice versa. Recall that a set is called relatively compact if it can be covered by a compact set. If such a map between Sx and S2 can be established, then each measure on Y represents a measure on X and vice versa. Such a map turns Y into a measure-representing model of X. Some examples will be given in chapter 2.

1.7 Metrics and convexity One of the concerns of contemporary integral geometry is the interrelation between the notions of metrics, convexity and measures in the spaces of lines and planes. In this and the next section we outline the simplest facts and leave more detailed discussion of this topic for chapter 5. Given a bounded convex domain D c 1R2 we define its breadth function b((p) to be the distance between the pair of parallel support lines of D which are orthogonal to the direction cp (see fig. 1.7.1; by definition, a support line has a point in common with dD but not with the interior of D). In general b(l9 0>2) = 0 if and only if ^ = ^ 2 ; (b) p(^ 1 ? ^ 3 ) ^ p ( ^ l 5