方差和重心矩的几何界限
Geometrical bounds for the variance and recentered moments
摘要：
我们根据随机向量的实现范围来约束它的方差和其他矩，从而将Popoviciu（1935）和Bhatia和Davis（2000）关于线上测度的不等式推广到多维。这是使用凸对偶和（无限维）线性规划。我们的边界的以下结果显示对称性破缺，提供了荣格定理（1901）的新证明，并证明了对吸引-排斥相互作用建模的聚集动力学的应用：在其支撑具有至多$\sqrt{2}$的直径的${\mathbf R}^n$上的概率测度中，我们表明，平均值周围的方差是最大化的那些措施，分配质量$1/(n+1)$的每个顶点的标准单形。对于$1 \le p <\infty$，$p$阶矩-最优居中-通过满足直径约束的那些中的相同措施来最大化。
英文摘要：
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (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.
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