论文标题
方差和近来时刻的几何界限
Geometrical bounds for the variance and recentered moments
论文作者
论文摘要
我们根据其实现范围绑定了随机向量的方差和其他力矩，从而将Popoviciu（1935）和Bhatia and Davis（2000）的不平等概括有关该线上的措施到几个维度。这是使用凸双重性和（无限维）线性编程完成的。 The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung's theorem (1901), and turns out to have applications to the aggregation dynamics modelling attractive-repulsive interactions: among probability measures on ${\mathbf R}^n$ whose support has diameter at most $\sqrt{2}$, we show that the variance around the mean is maximized precisely by those为标准单纯形的每个顶点分配质量$ 1/（n+1）$的度量。 对于$ 1 \ le p <\ infty $，$ p $ -th时刻---最佳居中---通过满足直径约束的人的相同度量的最大化。
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) 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 (1901), and turns out to have applications to the aggregation dynamics modelling attractive-repulsive interactions: among probability measures on ${\mathbf R}^n$ whose support has diameter at most $\sqrt{2}$, we show that the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a standard simplex. For $1 \le p <\infty$, the $p$-th moment --- optimally centered --- is maximized by the same measures among those satisfying the diameter constraint.
