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Necessary and Sufficient Convexity Conditions for the Ranges of Vector Measures

✍ Scribed by Rudolf Herschbach

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 686 KB
Volume: 181
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.3211810108

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✦ Synopsis

The absence of atoms in Lyapunov's Convexity Theorem is a sufficient, but not a necessary condition for the convexity of the range of an n-dimensional vector memure. In this paper algebraic and topological convexity conditions generalizing Lyapunov's Theorem are developed which are sufficient and necessary as well. Rom thew results the convene of Lyapunov's Theorem is derived in the form of a nonconvexity statement which @ves imight into the geometric structure of the ranges of vector measures with atoms. Anther, a characterization of the one-dimensional facea of a ronoid 2 , is given with respect to the generating spherical Bore1 mepsure p. As an appiication, it ia shown that the abeence of p -a t o m is a necessary and sufficient convexity condition for the range of the indefinite integral z dp, where z denotes the identical function on 9"''.

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