Introduction to Geometric Probability

by Daniel A. Klain, Gian-Carlo Rota

Overview

Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems.

Product Details

ISBN-13:	9780521593625
Publisher:	Cambridge University Press
Publication date:	12/11/1997
Series:	Lezioni Lincee
Pages:	196
Product dimensions:	5.51(w) x 8.50(h) x 0.55(d)

Table of Contents

Introduction; 1. The Buffon needle problem; 2. Valuation and integral; 3. A discrete lattice; 4. The intrinsic volumes for parallelotopes; 5. The lattice of polyconvex sets; 6. Invariant measures on Grassmannians; 7. The intrinsic volumes for polyconvex sets; 8. A characterization theorem for volume; 9. Hadwiger's characterization theorem; 10. Kinematic formulas for polyconvex sets; 11. Polyconvex sets in the sphere; References; Index of symbols; Index.

Editorial Reviews

"Elementary methods and exceptionally clear exposition bring a once seemingly advanced subject within the ken of a wide audience of mathematics students. Highly recommended for upper-division undergraduate and graduate students." Choice

"The exposition is marvellous: clear and precise... The powerful theory of valuations, intrinsic volumes and invariant measures built by Hadwiger, Groemer, McMullen and others is an impressive development. The beautiful exposition would make this volume worthwhile even if Klain and Rota hadn't 'something new' to say." Bulletin of the AMS

"The text is very elegant...This book is a very tantalizing one in that there is a definite sense that much of the subject is mature, even the combinatorial analogies." Mathematical Reviews

From the Publisher