Linear Perturbations of Quasiconvex Functions and Convexity

TL;DR

This paper characterizes convex functions on convex sets in real vector spaces through their linear perturbations, showing that certain quasiconvexity conditions imply convexity under mild stability assumptions.

Contribution

It provides a new characterization of convexity via linear perturbations and quasiconvexity, extending understanding of convex functions in vector spaces.

Findings

Convexity is equivalent to quasiconvexity of linear perturbations.

A mild stability property at boundary points suffices for the characterization.

The result applies to functions with specific boundary behavior on convex sets.

Abstract

Let $E$ be a real vector space with dual space $E^*$ and let $C\subset E$ be a convex subset with more than one point. Let $f : C\to\mathbb{R}$ be a function satisfying a mild stability property at 'flat' points of the (relative) boundary of $C$. We show that $f$ is convex if and only if for some linear form $c^*$ on $E$ not constant on $C$, the function $f+\lambda c^*$ is quasiconvex for all $\lambda\in\mathbb{R}$.