  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.
