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Journal of Convex Analysis 28 (2021), No. 1, 055--066
Copyright Heldermann Verlag 2021

Conjugate Convex Functions without Infinity

Fumioki Wada
Hokkaido Sapporo Kita Senior High School, Sapporo, Hokkaido 001-0025, Japan


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

Keywords: Conjugate function, convex function, Legendre-Fenchel transform, Lipschitz continuity, Fenchel inequality, subdifferential.

MSC: 46N10.

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