Volume of the polar of random sets and shadow systems
Metric Geometry
2013-11-18 v1 Functional Analysis
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-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities.
Keywords
Cite
@article{arxiv.1311.3690,
title = {Volume of the polar of random sets and shadow systems},
author = {Dario Cordero-Erausquin and Matthieu Fradelizi and Grigoris Paouris and Peter Pivovarov},
journal= {arXiv preprint arXiv:1311.3690},
year = {2013}
}