Randomized Urysohn-type inequalities
Probability
2019-10-28 v1
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.
Cite
@article{arxiv.1910.11654,
title = {Randomized Urysohn-type inequalities},
author = {Thomas Hack and Peter Pivovarov},
journal= {arXiv preprint arXiv:1910.11654},
year = {2019}
}