Contemporary Mathematics

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.

1.

Introduction

Lyapounov's celebrated convexity theorem of 1940 (e.g. [3, 10, 14, 15])

as-

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

Wolfowitz

[6]

says that the same is true for the

matrix-k-range

and the

partition

range

(see Definition 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 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

convex.

A different approach was adopted by Elton and Hill [7], 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

of comparison.

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

1

http://dx.doi.org/10.1090/conm/234/03441