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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convex bodies and concave functions
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by M. Meyer, G. Mokobodzki and M. Rogalski PDF
Proc. Amer. Math. Soc. 123 (1995), 477-484 Request permission

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 \[ 0 \leq f(t) \leq p(K(t)),\quad for\;every\;t \in [0,1].\] Then there exists a closed convex subset L of $[0,1] \times E$ such that $L \subset K$, \[ L(t) = \{ z \in E;(t,z) \in L\} \in \mathfrak {C}\quad and\quad f(t) = p(L(t)) for\;every\;t \in [0,1].\] Some examples and applications are given to the study of Steiner symmetrization and of the Riesz decomposition property for concave continuous functions.
References
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 477-484
  • MSC: Primary 52A21; Secondary 46B99, 52A05
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1254848-2
  • MathSciNet review: 1254848