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

Authors: John Elton and Theodore P. Hill
Journal: Proc. Amer. Math. Soc. 99 (1987), 297-304
MSC: Primary 28B05; Secondary 46G10, 49B36, 60A10
MathSciNet review: 870789
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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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Keywords: Range of a vector measure, convexity theorem, vector measures with atoms, dent-size of a nonconvex set, zonotope, zonohedron
Article copyright: © Copyright 1987 American Mathematical Society

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