Summary: A SHARP NON-CONVEXITY BOUND FOR PARTITION
RANGES OF VECTOR MEASURES WITH ATOMS
PIETER C. ALLAART
University of North Texas, Denton, TX 76203
Abstract. A sharp upper bound is given for the degree of non-convexity of
the partition range of a nite-dimensional vector measure, in terms of the
maximum (one-dimensional) mass of the atoms of that measure. This upper
bound improves on a bound of Hill and Tong (1989) by an order of magnitude
pn. Its proof uses several ideas from graph theory, combinatorics, and convex
geometry. Applications are given to optimal-partitioning and fair division
Date: March 4, 1999.
1991 Mathematics Subject Classi cation. Primary 28B05, 51K99, 52B12 Secondary 05C20,
Key words and phrases. Partition range, optimal-partitioning, convexity theorem, vector mea-
sure, vector atom, Hausdor -distance, digraph, tree.
A Sharp Non-convexity Bound
Pieter C. Allaart