 
Summary: Contemporary Mathematics
Bounds on the Nonconvexity of Ranges
of Vector Measures with Atoms
Pieter C. Allaart
Abstract. Upper bounds are given for the distance between the range, matrix
range and partition range of a vector measure to the respective convex hulls
of these ranges. The bounds are speci ed in terms of the maximum atom size,
and generalize convexity results of Lyapounov (1940) and Dvoretzky, Wald
and Wolfowitz (1951). Applications are given to the bisection problem, the
"problem of the Nile", and fair division problems.
1. Introduction
Lyapounov's celebrated convexity theorem of 1940 (e.g. 3, 10, 14, 15]) as
serts that the range of a nitedimensional, atomless vector measure is convex and
compact. A generalization of Lyapounov's theorem due to Dvoretzky, Wald and
Wolfowitz 6] says that the same is true for the matrixkrange and the partition
range (see De nition 2.2 below).
If the vector measure has atoms, then convexity of all three ranges may fail
in general, although atomlessness is not a necessary condition. Gouweleeuw 9]
has given necessary and su cient conditions for the range (or matrixkrange) to
be convex, as well as nontrivial su cient conditions for the partition range to be
