Introduction to Geometric Probability
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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Aff(n antichain Buffon needle problem Chapter compact convex sets compute continuous invariant valuation convex body denote discrete distributive lattice elements equation Euler characteristic exists facets finite unions flag coefficients Flag(n follows geometric probability given Gr(n Graff(n Grassmannians Haar Hadwiger's characterization Theorem Helly's theorem hyperplane implies inclusion-exclusion inclusion-exclusion principle indicator functions induction integral intrinsic volumes invariant measure invariant simple valuation L.Y.M. inequality Lemma line segment linear variety maximal element mean projection formula Minkowski sum Mod(n non-empty interiors normalization Note number of intersections obtain order ideals orthogonal Par(n parallelotopes partially ordered sets permutation plane points polyconvex sets positive integers principal kinematic formula Proof Let proof of Theorem Proposition random Recall rigid motion Rn+1 rotation Section set function simplex simplicial complex Sperner's theorem star-shaped sets straight line subset subspaces Suppose surface area symmetric theory unique unit ball valuation defined vanishes variable vector space