A Generalization of Lyapounov\u27s Convexity Theorem to Measures with Atoms

摘要

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\u27s Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.