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Journal of Convex Analysis 29 (2022), No. 3, 929--937
Copyright Heldermann Verlag 2022

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

Thomas Ruf
Institut für Mathematik, Universität Augsburg, Augsburg, Germany

Bernd Schmidt
Institut für Mathematik, Universität Augsburg, Augsburg, Germany


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\\[2mm] \centerline{$\displaystyle \min_{x \in X} f(x) - \langle x' , x \rangle_X$}\\[-2mm] 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.

Keywords: Locally convex Hausdorff space, (essentially) strictly convex function, biconjugate, convex envelope, convex hull, subdifferential, uniqueness.

MSC: 46G05, 52A07, 46N10, 49N15.

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