Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [A] Alano Ancona, Sur les espaces de Dirichlet: principes, fonction de Green, J. Math. Pures Appl. (9) 54 (1975), 75–124 (French). MR 0457756 (56 #15960)
  • [B-Z] 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 (89b:52020)
  • [F] William J. Firey, A functional characterization of certain mixed volumes, Israel J. Math. 24 (1976), no. 3-4, 274–281. MR 0640871 (58 #30744)
  • [M-S] 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


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1254848-2
PII: S 0002-9939(1995)1254848-2
Keywords: Concave functions, convex bodies
Article copyright: © Copyright 1995 American Mathematical Society