Introduction to Geometric Probability

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Cambridge University Press, Dec 11, 1997 - Mathematics - 178 pages
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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.
  

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Contents

A discrete lattice
13
The intrinsic volumes for parallelotopes
30
Invariant measures on Grassmannians
60
The intrinsic volumes for polyconvex sets
86
Hadwigers characterization theorem
118
Kinematic formulas for polyconvex sets
146
Polyconvex sets in the sphere
154
Bibliography
168
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