Extendability of continuous quasiconvex functions from subspaces

TL;DR

This paper investigates the extendability of continuous quasiconvex functions from subspaces in topological vector spaces, establishing conditions under which such functions can be extended while preserving quasiconvexity, and generalizing previous finite-dimensional results.

Contribution

It introduces and analyzes the properties $(QE)$ and $(CE)$, proving their implications and equivalences under certain conditions, and extends known extension results to infinite-dimensional separable spaces.

Findings

$(QE)$ always implies $(CE)$.

Under certain conditions, $(QE)$ and $(CE)$ are equivalent.

Generalizes extension results to infinite-dimensional separable spaces.

Abstract

Let $Y$ be a subspace of a topological vector space $X$, and $A\subset X$ an open convex set that intersects $Y$. We say that the property $(QE)$ [property $(CE)$] holds if every continuous quasiconvex [continuous convex] function on $A\cap Y$ admits a continuous quasiconvex [continuous convex] extension defined on $A$. We study relations between $(QE)$ and $(CE)$ properties, proving that $(QE)$ always implies $(CE)$ and that, under suitable hypotheses (satisfied for example if $X$ is a normed space and $Y$ is a closed subspace of $X$), the two properties are equivalent. By combining the previous implications between $(QE)$ and $(CE)$ properties with known results about the property $(CE)$, we obtain some new positive results about the extension of quasiconvex continuous functions. In particular, we generalize the results contained in \cite{DEQEX} to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which $(QE)$ does not hold.