# Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

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 $$f$$?