Randomized Urysohn–type Inequalities

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.

The Petty projection inequality for sets of finite perimeter

“…F. Gao, D. Hug and R. Schneider [12] proved the Urysohn inequality and the Blaschke-Santaló inequality in the spherical and hyperbolic space. Later, T. Hack and P. Pivovarov [14] proved a randomized version of the spherical and hyperbolic Urysohn-type inequalities. G. Wang and C. Xia [25] solved various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space H n and established quite strong Alexandrov-Fenchel type inequalities.…”

“…In comparison with the Euclidean case there are very few results and techniques available in spherical and hyperbolic space. In [12] and [14], the authors use the two-point symmetrization procedure together with rearrangement inequalities in order to prove their result. In [2], the authors used a probabilistic approach to prove the spherical centroid inequality.…”