Volume 234, 1999
Bounds on the non-convexity 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 specified 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.
Lyapounov's celebrated convexity theorem of 1940 (e.g. [3, 10, 14, 15])
serts that the range of a finite-dimensional, atomless vector measure is convex and
compact. A generalization of Lyapounov's theorem due to Dvoretzky, Wald and
says that the same is true for the
(see Definition 2.2 below).
the vector measure has atoms, then convexity of all three ranges may fail
in general, although atomlessness is not a necessary condition. Gouweleeuw
has given necessary and sufficient conditions for the range (or matrix-k-range) to
be convex, as well as non-trivial sufficient conditions for the partition range to be
A different approach was adopted by Elton and Hill , who proved a bound
on how far from convex the range may be, as a function of the maximum atom size.
The aim of this paper is to present such non-convexity inequalities for the three
types of ranges mentioned above. Some of these are sharp, whereas in other cases
the best possible bounds are not known to the author.
The first result is a slightly improved, but sharp, version of Elton and Hill's
inequality. The proof presented here is very similar to that of Elton and Hill, with
only a few minor adaptations. The original inequality is also included for the sake
Next in line are two non-convexity inequalities for the matrix-k-range. These
are proved using the improved inequality for the range, and a device of chaining
1991 Mathematics Subject Classification. Primary 28805; Secondary 60A10.
Key words and phmses. Range of a vector measure, partition range, matrix range, vector
atom, convexity theorems, Hausdorff distance.
1999 American Mathematical Society