Can someone please clarify what is exact reason why $L$-Lipschitz continuity of gradient of convex function $f$ may not imply $1/L$-strong convexity of its Fenchel conjugate $f^*$ in the case when $f$ is defined on $S \subsetneq \mathbb{R}^d$? For example, if $f(x) = +\infty$ for all $x \notin S$.
2026-02-23 20:42:27.1771879347
Why Lipscitz smoothness not always implies strong convexity of Fenchel conjugate?
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