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
thomas.ruf@math.uni-augsburg.de
Bernd Schmidt
Institut für Mathematik, Universität Augsburg, Augsburg, Germany
bernd.schmidt@math.uni-augsburg.de
[Abstract-pdf]
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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