A Sharp Nonconvexity Bound for Partition Ranges of Vector Measures with Atoms

A sharp upper bound is given for the degree of nonconvexity of the partition range of a finite-dimensional vector measure, in terms of the maximum one-di- mensional. mass of the atoms of that measure. This upper bound improves on a bound of Hill and Tong 1989, Anal. Stat. 17, 1325]1334.by an order of magnitude ʹn . Its proof uses several ideas from graph theory, combinatorics, and convex geometry. Applications are given to optimal-partitioning and fair division problems.