Continuous Selections of Lipschitz Extensions in Metric Spaces

TL;DR

This paper investigates the continuity properties of Lipschitz extension operators in metric spaces with curvature bounds, demonstrating conditions under which continuous selections exist and extensions can be made with controlled Lipschitz constants.

Contribution

It establishes lower semi-continuity and the existence of continuous selections for multivalued Lipschitz extension operators under curvature assumptions.

Findings

Extension operators are lower semi-continuous with continuous selections.

Lipschitz mappings can be extended with images in the closure of convex hulls.

In hyperconvex spaces, extensions can be nonexpansive.

Abstract

This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the closure of the convex hull of the image of the original mapping. When the target space is hyperconvex one can obtain in fact nonexpansivity.