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.

中文翻译：

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随机 Urysohn 型不等式

作为欧几里德空间中 Urysohn 不等式的自然类比，Gao、Hug 和 Schneider 在 2003 年表明，在球面或双曲空间中，当 K 是测地球时，满足给定凸体 K 的完全测地超曲面的总度量最小化。我们通过将 K 取为根据概率分布绘制的有限多个点的凸包，并通过显示测地线球上的均匀分布达到最小值来呈现该结果的随机扩展。作为推论，我们获得了球体上的随机 Blaschke-Santalo 不等式。