 
Summary: A SHARP NONCONVEXITY 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 nonconvexity of
the partition range of a nitedimensional vector measure, in terms of the
maximum (onedimensional) 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 optimalpartitioning and fair division
problems.
Date: March 4, 1999.
1991 Mathematics Subject Classi cation. Primary 28B05, 51K99, 52B12 Secondary 05C20,
60A10, 90A06.
Key words and phrases. Partition range, optimalpartitioning, convexity theorem, vector mea
sure, vector atom, Hausdor distance, digraph, tree.
1
A Sharp Nonconvexity Bound
Author's address:
Pieter C. Allaart
