Random ball-polyhedra and inequalities for intrinsic volumes
Abstract
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.
Keywords
Cite
@article{arxiv.1510.07292,
title = {Random ball-polyhedra and inequalities for intrinsic volumes},
author = {Grigoris Paouris and Peter Pivovarov},
journal= {arXiv preprint arXiv:1510.07292},
year = {2016}
}
Comments
Sections 4 and 5 have been rewritten, along with the relevant motivation for these in the introduction. We now compare our results with kindred work on intersections of random halfspaces. We also discuss further consequences of the stochastic dominance in Theorem 1.1