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Necessary and Sufficient Convexity Conditions for the Ranges of Vector Measures
Author(s) -
Herschbach Rudolf
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211810108
Subject(s) - convexity , mathematics , measure (data warehouse) , converse , range (aeronautics) , lyapunov function , pure mathematics , algebraic number , mathematical analysis , nonlinear system , geometry , materials science , physics , database , quantum mechanics , computer science , financial economics , economics , composite material
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 measure. In this paper algebraic and topological convexity conditions generalizing Lyapunov's Theorem are developed which are sufficient and necessary as well. From these results the converse of Lyapunov's Theorem is derived in the form of a nonconvexity statement which gives insight into the geometric structure of the ranges of vector measures with atoms. Further, a characterization of the one‐dimensional faces of a zonoid Z μ , is given with respect to the generating spherical Borel measure μ. As an application, it is shown that the absence of μ ‐ atoms is a necessary and sufficient convexity condition for the range of the indefinite integral ∫ x dμ where x denotes the identical function on S n‐1 .