Random ball-polyhedra and inequalities for intrinsic volumes

TL;DR

This paper introduces a randomized approach to inequalities involving intrinsic volumes of convex bodies, using stochastic approximation and new isoperimetric inequalities, with implications for extremal measures.

Contribution

It develops a stochastic approximation method for convex bodies and establishes new isoperimetric inequalities for intrinsic volumes, extending classical geometric inequalities.

Findings

Extends Urysohn inequality using randomized methods.

Identifies extremizers as uniform measures on Euclidean ball or cube.

Provides new isoperimetric inequalities for intersections of balls.

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.