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

DOI:
https://doi.org/10.1090/S0002-9939-1995-1254848-2

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

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Keywords:
Concave functions,
convex bodies

Article copyright:
© Copyright 1995
American Mathematical Society