Extending quasiconvex functions from uniformly convex sets

TL;DR

This paper investigates the extension of Lipschitz quasiconvex functions from convex subsets of finite-dimensional normed spaces to the entire space, revealing limitations for Lipschitz extensions and characterizing conditions for continuous extensions based on geometric properties.

Contribution

It establishes that Lipschitz quasiconvex functions cannot generally be extended Lipschitzly, and provides geometric criteria for continuous extendability.

Findings

Lipschitz extensions are generally impossible for quasiconvex functions.

Continuous extensions depend on specific geometric properties of the set.

Characterization of extendability conditions based on the geometry of the convex set.

Abstract

Let $X$ be a normed space of a finite dimension at least two, and $C\subsetneq X$ a closed convex set with nonempty interior. We are interested in extending Lipschitz quasiconvex functions on $C$ to quasiconvex functions on $X$. We show that, unlike what holds for convex functions, in general one cannot obtain Lipschitz extensions (except for trivial cases). If we require just uniformly continuous or continuous extensions, such extendability properties for $C$ are shown to be characterized by some geometric properties of $C$.