It is trivial to see that the Fenchel conjugate/dual given by
$$f^*(y) = \sup\limits_{x \in X}\{x^Ty - f(x)\}$$ of any function $f$ is convex.
Is it true (as a necessarily and sufficient condition) that if $f$ is strictly concave, then $f^*$ is strictly convex?
Are there less restrictive conditions under which the dual $f^*$ is strictly convex?