Introduction to Geometric ProbabilityHere 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. |
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 |
| 168 | |
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Common terms and phrases
a₁ Aff(n antichain B₁ Chapter compact convex sets compute containing continuous invariant valuation convex body convex polytope coordinate denote discrete distributive lattice E(XK elements Euler characteristic exists finite unions flag coefficients follows geometric probability given Gr(n Graff(n,k Grassmannians H₁ Hadwiger's characterization Theorem Helly's theorem hyperplane implies inclusion-exclusion inclusion-exclusion principle indicator functions integral intrinsic volumes invariant measure invariant simple valuation K₁ L.Y.M. inequality line segment linear variety maximal element mean projection formula Minkowski sum Mod(n Mult(n normalization Note order ideals orthogonal P₁ Par(n parallelotopes partially ordered sets permutation plane Polycon(n polyconvex sets positive integers principal kinematic formula Proof Let Proposition Recall relint(Q rigid motion rotation s-system Section set function simplex simplicial complex SO(n Sperner's theorem spherical straight line subset subspaces Suppose symmetric unique unit ball valuation defined μη μκ μο