# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

by M. Meyer, G. Mokobodzki and M. Rogalski
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
• Alano Ancona, Sur les espaces de Dirichlet: principes, fonction de Green, J. Math. Pures Appl. (9) 54 (1975), 75–124 (French). MR 457756
• Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
• William J. Firey, A functional characterization of certain mixed volumes, Israel J. Math. 24 (1976), no. 3-4, 274–281. MR 640871, DOI 10.1007/BF02834758
• G. Mokobodzki and D. Sibony, Sur une propriété caractéristique des cônes de potentiels, C. R. Acad. Sci. Paris Sér. I Math. 266 (1968), 215-218.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A21, 46B99, 52A05
• Retrieve articles in all journals with MSC: 52A21, 46B99, 52A05