A generalization of Lyapounov’s convexity theorem to measures with atoms

Elton, John; Hill, Theodore P.

doi:10.1090/S0002-9939-1987-0870789-X

A generalization of Lyapounov’s convexity theorem to measures with atoms
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by John Elton and Theodore P. Hill PDF

Proc. Amer. Math. Soc. 99 (1987), 297-304 Request permission

Abstract:

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 $\alpha n/2$, where $\alpha$ is the largest (one-dimensional) mass of the atoms of the measure. The case $\alpha = 0$ yields Lyapounov’s Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.

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