Convex values and Lipschitz behavior of the complete hull mapping

Abstract:

This note continues the study initiated in 2006 by P.L. Papini, R. R. Phelps and the author on some classical notions from finite-dimensional convex geometry in spaces of continuous functions. Let $\mathcal H$ be the family of all closed, convex and bounded subsets of a Banach space endowed with the Hausdorff metric. A completion of $A\in \mathcal H$ is a diametrically maximal set $D\in \mathcal H$ satisfying $A\subset D$ and $\operatorname {diam} A=\operatorname {diam} D$. The complete hull mapping associates with every $A\in \mathcal H$ the family $\gamma (A)$ of all its possible completions. It is shown that the set-valued mapping $\gamma$ need not be convex valued even in finite-dimensional spaces, while, in the case of $C(K)$ spaces, $\gamma$ is convex valued if and only if $K$ is extremally disconnected. Regarding the continuity we prove that, again in $C(K)$ spaces, $\gamma$ is always Lipschitz continuous with constant less than or equal to 5 and has a Lipschitz selection with constant less than or equal to 3. If we consider the analogous problem in Euclidean spaces, we show that $\gamma$ is Hölder continuous of order 1/4 and locally Hölder continuous of order 1/2, the Hölder constants depending on the diameter of the sets in both cases.