On continuous choice of retractions onto nonconvex subsets

TL;DR

This paper establishes conditions under which continuous retractions onto nonconvex subsets of Banach and Hilbert spaces can be selected, with the nonconvexity measure constrained below a specific threshold.

Contribution

It proves the existence of continuous retractions onto nonconvex sets with controlled nonconvexity, extending previous geometric results to Banach and Hilbert spaces.

Findings

Retractions depend continuously on the set when nonconvexity is less than 1/2.

The set of all uniform retractions onto an α-paraconvex set is itself α/(1-α)-paraconvex.

In Hilbert spaces, the nonconvexity bound can be improved and the threshold reduced.

Abstract

For a Banach space $B$ and for a class $\A$ of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements $A \in \A$ can be chosen to depend continuously on $A$, whenever nonconvexity of each $A \in \A$ is less than $\f{1}{2}$. The key geometric argument is that the set of all uniform retractions onto an $\a-$paraconvex set (in the spirit of E. Michael) is $\frac{\a}{1-\a}-$paraconvex subset in the space of continuous mappings of $B$ into itself. For a Hilbert space $H$ the estimate $\frac{\a}{1-\a}$ can be improved to $\frac{\a (1+\a^{2})}{1-\a^{2}}$ and the constant $\f{1}{2}$ can be reduced to the root of the equation $\a+ \a^{2}+a^{3}=1$.