Abstract

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let XX be a locally convex Hausdorff space and f ⁣:X→(−∞,∞]f\colon X\to (-\infty, \infty] a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem
min⁡x∈Xf(x)−⟨x′,x⟩X\displaystyle \min_{x \in X} f(x) - \langle x', x \rangle_X
admits at most one minimizer as x′x' ranges over dom (∂f∗)\text{\rm dom}\, \left( \partial f^* \right) if and only if the biconjugate f∗∗f^{**} is essentially strictly convex and agrees with ff at all points where f∗∗f^{**} is subdifferentiable. We prove this via a representation formula for f∗∗f^{**} that might be of independent interest.

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Keywords

  • Locally convex Hausdorff space
  • (essentially) strictly convex function
  • biconjugate
  • convex envelope
  • convex hull
  • subdifferential
  • uniqueness

Mathematics Subject Classification

46G05, 52A07, 46N10, 49N15

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Suggested citation

T. Ruf, B. Schmidt. “Unique Minimizers and the Representation of Convex Envelopes in Locally Convex Vector Spaces.” Journal of Convex Analysis 29 (2022), No. 3, 929–937. https://doi.org/10.68381/jca29053

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.