
Journal of Convex Analysis 28 (2021), No. 1, 055066 Copyright Heldermann Verlag 2021 Conjugate Convex Functions without Infinity Fumioki Wada Hokkaido Sapporo Kita Senior High School, Sapporo, Hokkaido 0010025, Japan 21013351@alumni.tus.ac.jp [Abstractpdf] 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 realvalued 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, LegendreFenchel transform, Lipschitz continuity, Fenchel inequality, subdifferential. MSC: 46N10. [ Fulltextpdf (110 KB)] for subscribers only. 