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Random Polytopes and Affine Surface Area

✍ Scribed by Carsten Schütt

Publisher
John Wiley and Sons
Year
2006
Tongue
English
Weight
686 KB
Volume
170
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let K be a convex body in R__^d^__. A random polytope is the convex hull [x~1~, …, x~n~] of finitely many points chosen at random in K. IE(K, n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with C^3^ boundary and everywhere positive curvature
where x(x) denotes the Gauß‐Kronecker curvature. We show that the same formula holds for all convex bodies if x(x) denotes the generalized Gauß‐Kronecker curvature.

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