Convex bodies and concave functions

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.