Published in Proceedings of the American Mathematical Society, Volume 99, Issue 2, February 1, 1987, pages 297-304. Copyright © 1987 American Mathematical Society. The definitive version is available at http://www.jstor.org/stable/2046629.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
The distance from the convex hull of the range of an n-dimensional vector-valued measure to the range of that measure is no more than α n/2, where α is the largest (one-dimensional) mass of the atoms of the measure. The case α = 0 yields Lyapounov's Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.