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

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