Introduction to Geometric Probability (Lezioni Lincee) [Hardcover]

Daniel A. Klain (Author), Gian-Carlo Rota (Author)

Book Description

Publication Date: December 13, 1997 | ISBN-10: 052159362X | ISBN-13: 978-0521593625

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.

Editorial Reviews

Review

"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

Book Description

Elementary geometry in n-dimensional Euclidean space is a subject that, under the stimulus of computational geometry, is regaining its former position. This is the first textbook that addresses some fundamental problems of Euclidean geometry that have been solved over the last half-century. The authors, who have made significant contributions to the subject, have taken pains to keep the exposition elementary, making the relationship between it and combinatorics transparent. It should be required reading of anyone in mathematics or computer science who deals with the visual display of information.

Product Details

• Hardcover: 196 pages
• Publisher: Cambridge University Press (December 13, 1997)
• Language: English
• ISBN-10: 052159362X
• ISBN-13: 978-0521593625
• Product Dimensions: 8.5 x 5.6 x 0.8 inches
• Shipping Weight: 13.4 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #4,490,197 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

8 of 8 people found the following review helpful:

5.0 out of 5 stars a beautiful book, August 11, 2000

By A Customer

This review is from: Introduction to Geometric Probability (Lezioni Lincee) (Hardcover)

Everyone with any interest in probability or combinatorics should take a look at this book, and at least read Chapter 1, on the Buffon needle problem. It gives a beautiful conceptual solution, quite different from the more well-known solution using integrals to get conditional probabilities. I find it hard to imagine anyone reading Chapter 1 and not deciding to read the entire book.

I heard Rota lecture on this material, and the book has much the same feeling as his lectures: it is clear, elegant, and concise, full of illuminating examples. Relatively little background is required, and it should be easily accessible to beginning graduate students (or undergraduates with unusually strong backgrounds).