Volume of the polar of random sets and shadow systems |
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| Authors: | Dario?Cordero-Erausquin Matthieu?Fradelizi Grigoris?Paouris Email author" target="_blank">Peter?PivovarovEmail author |
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| Institution: | 1.Institut de Mathématiques de Jussieu-PRG,Université Pierre et Marie Curie (Paris 6),Paris Cedex 05,France;2.Laboratoire d’Analyse et de Mathématiques Appliquées,Université Paris-Est Marne la Vallée,Marne la Vallée Cedex 2,France;3.Department of Mathematics (Mailstop 3368),Texas A&M University,College Station,USA;4.Mathematics Department,University of Missouri,Columbia,USA |
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| Abstract: | We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke–Santaló inequality which, in turn, can be derived by the law of large numbers. The method involves shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities. |
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