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It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual function $g$ which has the same properties, whereby the gradient is a bijection from the domain of $f$ to the domain of $g$.

Does there exist a similar set of results for strongly convex $f$? or for essentially strongly convex $f$?

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Yes, strong convexity is conjugate to uniform smoothness or Lipschitz-continuous differentiability (where the Lipschitz constant is the reciprocal of the modulus of strong continuity), see, e.g.,

Azé, Dominique; Penot, Jean-Paul, Uniformly convex and uniformly smooth convex functions, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 4, No. 4, 705-730 (1995). ZBL0870.49010.

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  • $\begingroup$ Does the gradient remain a bijection? If not, what guarantees it? $\endgroup$
    – Concu Bine
    Commented Sep 6, 2019 at 1:37
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    $\begingroup$ Yes, strongly convex functions are in particular strictly convex, which is the property that guarantees the bijection (of the subgradient as a set-valued operator, which reduces to the gradient if you also have smoothness). $\endgroup$
    – Christian Clason
    Commented Sep 6, 2019 at 6:34

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