On Strong Quasiconvexity of Functions in Infinite Dimensions

TL;DR

This paper investigates the properties of $\sigma$-quasiconvex functions in infinite-dimensional normed spaces, extending known results and providing new insights into their structure, coercivity, and minimization behavior.

Contribution

It introduces new results on strong quasiconvexity of functions in infinite dimensions and extends finite-dimensional properties to this broader setting.

Findings

Preservation of $\sigma$-quasiconvexity under certain operations

New results on strong quasiconvexity of norm and Minkowski functions in infinite dimensions

Extension of supercoercivity results to infinite-dimensional spaces

Abstract

In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start by analyzing certain operations on functions that preserve $\sigma$-quasiconvexity. Next, we present new results concerning the strong quasiconvexity of norm and Minkowski functions in infinite dimensions. Furthermore, we extend a recent result by F. Lara [16] on the supercoercive properties of strongly quasiconvex functions, with applications to the existence and uniqueness of minima, from finite dimensions to infinite dimensions. Finally, we address counterexamples related to strong quasiconvexity.