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

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 $f\colon X\to (-\infty, \infty]$ a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem
$\displaystyle \min_{x \in X} f(x) - \langle x', x \rangle_X$
admits at most one minimizer as $x'$ ranges over $\text{\rm dom}\, \left( \partial f^* \right)$ 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.