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

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 $f:X\mapsto \left(-\infty ,\infty \right]$ 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 $x^{\prime }$ ranges over $\text{dom}\left(\partial f^{\ast }\right)$ if and only if the biconjugate $f^{\ast \ast }$ is essentially strictly convex and agrees with $f$ at all points where $f^{\ast \ast }$ is subdifferentiable. We prove this via a representation formula for $f^{\ast \ast }$ that might be of independent interest.