Compact Hölder retractions and nearest point maps

Abstract

In this paper, two main results concerning uniformly continuous retractions are proved. First, an $\alpha$-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive $\alpha <1$. This constitutes a positive solution to a Hölder version of a question raised by Godefroy and Ozawa. In fact, compact convex sets are found to be absolute $\alpha$-Hölder retracts under certain assumption of flatness. Second, we provide an example of a strictly convex Banach space $X$ arbitrarily close to ${\ell }_2$ (for the Banach Mazur distance) and a finite dimensional compact convex subset of $X$ for which the nearest point map is not uniformly continuous even when restricted to bounded sets.

21 pages