Compact H\"older retractions and nearest point maps

TL;DR

This paper constructs H"older retractions onto compact convex sets in Banach spaces and explores the limitations of nearest point maps, revealing new insights into the structure of Banach spaces.

Contribution

It introduces a method to construct H"older retractions in Banach spaces and provides counterexamples for the uniform continuity of nearest point maps.

Findings

Existence of $eta$-H"older retractions for all positive $eta<1$

Compact convex sets can be absolute $eta$-H"older retracts under flatness conditions

Counterexample showing nearest point map not uniformly continuous in certain Banach spaces

Abstract

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.