Compact H\"older retractions and nearest point maps

In this paper, two main results concerning uniformly continuous retractions are proved. First, an $\alpha$-H\"older 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 H\"older version of a question raised by Godefroy and Ozawa. In fact, compact convex sets are found to be absolute $\alpha$-H\"older 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.