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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 $x \notin S$.

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  • $\begingroup$ If you assume that the conjugate is strongly convex, which contradicyion can you derive ? $\endgroup$
    – dohmatob
    Oct 21, 2023 at 13:24

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