Unique Minimizers and the Representation of Convex Envelopes in Locally Convex Vector Spaces

TL;DR

This paper characterizes when a convex envelope in locally convex spaces has unique minimizers, linking this to the essential strict convexity of the biconjugate and providing a new representation formula.

Contribution

It establishes a partial converse to known convexity results, connecting unique minimizers to the properties of the convex envelope and introduces a novel representation formula for the biconjugate.

Findings

Unique minimizers correspond to the essential strict convexity of the biconjugate.

The biconjugate $f^{**}$ matches $f$ where it is subdifferentiable.

A new representation formula for $f^{**}$ is developed.

Abstract

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem \begin{equation} \label{eq. minimization problem} \min_{x \in X} f(x) - \langle x' , x \rangle_X \end{equation} admits at most one minimizer as $x'$ ranges over $\text{dom} \left( \partial f^* \right)$ if and only if the biconjugate $f^{**}$ is essentially strictly convex and agrees with $f$ at all points where $f^{**}$ is subdifferentiable. We prove this via a representation formula for $f^{**}$ that might be of independent interest.