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 $f$? 



Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?
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**](http://dx.doi.org/10.5802/afst.809), Ann. Fac. Sci. Toulouse, VI. Sér., Math. 4, No. 4, 705-730 (1995). [ZBL0870.49010](https://zbmath.org/?q=an:0870.49010).


