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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convex bodies and concave functions

Authors: M. Meyer, G. Mokobodzki and M. Rogalski
Journal: Proc. Amer. Math. Soc. 123 (1995), 477-484
MSC: Primary 52A21; Secondary 46B99, 52A05
MathSciNet review: 1254848
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Abstract: We find properties that a class $ \mathfrak{C}$ of closed bounded convex subsets of a Banach space E and a mapping $ p:\mathfrak{C} \to {\mathbb{R}_ + }$ should satisfy in order to obtain the following result:

Theorem. Let $ \mathfrak{C}$ and $ p:\mathfrak{C} \to {\mathbb{R}_ + }$ satisfy these properties, and let K be a closed convex subset of $ [0,1] \times E$ such that for every $ t \in [0,1]$ the set $ K(t) = \{ z \in E;(t,z) \in K\} $ is an element of $ \mathfrak{C}$. Suppose that a concave continuous function $ f:[0,1] \to \mathbb{R}$ is given such that

$\displaystyle 0 \leq f(t) \leq p(K(t)),\quad for\;every\;t \in [0,1].$

Then there exists a closed convex subset L of $ [0,1] \times E$ such that $ L \subset K$,

$\displaystyle L(t) = \{ z \in E;(t,z) \in L\} \in \mathfrak{C}\quad and\quad f(t) = p(L(t)) for\;every\;t \in [0,1].$

Some examples and applications are given to the study of Steiner symmetrization and of the Riesz decomposition property for concave continuous functions.

References [Enhancements On Off] (What's this?)

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Keywords: Concave functions, convex bodies
Article copyright: © Copyright 1995 American Mathematical Society

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