Randomized Urysohn-type inequalities

TL;DR

This paper extends Urysohn-type inequalities to a randomized setting on spherical and hyperbolic spaces, showing that convex hulls of randomly chosen points minimize certain measures when uniformly distributed on geodesic balls.

Contribution

It introduces a probabilistic version of Urysohn inequalities, demonstrating that convex hulls of random points minimize total measure of geodesic hypersurfaces, with implications for a randomized Blaschke--Santalo inequality.

Findings

Convex hulls of random points minimize geodesic hypersurface measures.

Uniform distributions on geodesic balls are optimal for minimization.

Results extend classical inequalities to a probabilistic framework.

Abstract

As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic hypersurfaces meeting a given convex body K is minimized when K is a geodesic ball. We present a random extension of this result by taking K to be the convex hull of finitely many points drawn according to a probability distribution and by showing that the minimum is attained for uniform distributions on geodesic balls. As a corollary, we obtain a randomized Blaschke--Santalo inequality on the sphere.