Abstract

Let Br(E)B_{r}(E) be the closed ball of radius rr around the origin in a real Banach space EE and Fr(E)\mathcal{F}_{r}(E) be the set of all rr-Lipschitz continuous convex functions defined on Br(E)B_{r}(E). Suppose ff is a real-valued and bounded below function on Br(E)B_{r}(E). We define the II-conjugate function fIf^{I} of ff to improve the Fenchel inequality and investigate the properties of fIf^{I}. In particular, (fI)I(f^{I})^{I} coincides with ff on Br(E)B_{r}(E) if and only if ff is in Fr(E)\mathcal{F}_{r}(E). Excluding the value +∞+\infty, the transformation from ff to fIf^{I} enlarges the potentiality of the contribution to numerical computation for convex analysis.

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Keywords

  • Conjugate function
  • convex function
  • Legendre-Fenchel transform
  • Lipschitz continuity
  • Fenchel inequality
  • subdifferential

Mathematics Subject Classification

46N10

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Suggested citation

F. Wada. “Conjugate Convex Functions without Infinity.” Journal of Convex Analysis 28 (2021), No. 1, 55–66. https://doi.org/10.68381/jca28006

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.