RANDOMIZED URYSOHN–TYPE INEQUALITIES. Issue 1 (23rd November 2020)

• Record Type: Journal Article
• Authors: Hack, Thomas; Pivovarov, Peter
• Is Part Of: Mathematika. Volume 67:Issue 1(2021)
• Page Start: 100
• Page End: 115
• Publication Date: 2020-11-23
• Subjects: 52A22 -- 52A40 -- 52A55 -- 60D05 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12063
• Languages: English
• ISSNs: 0025-5793

Abstract

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–Santaló inequality on the sphere.