Convex values and Lipschitz behavior of the complete hull mapping

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.