Geometrical Bounds for Variance and Recentered Moments

• 成果类型: Article
• 署名作者: Lim, Tongseok; McCann, Robert J.
• 署名单位: Purdue University System; Purdue University; University of Toronto
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1125
• 发表日期: 2022
• 页码: 286-296
• 关键词: radial symmetry; sphere; minimizers; convexity; MODEL

摘要

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu and of Bhatia and Davis concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung's theorem, and turns out to have applications to the aggregation dynamics modelling attractive-repulsive interactions: among probability measures on R-n whose support has diameter at most root 2, we show that the variance around the mean is maximized precisely by those measures that assign mass 1/(n + 1) to each vertex of a standard simplex. For 1 <= p < infinity, the p th moment-optimally centered-is maximized by the same measures among those satisfying the diameter constraint.