True or False: Fenchel dual $f^*$ is strictly convex if and only if $f$ is strictly concave?

Question

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?

Answer 1

The answer is No in general!

Take $f(x)= - \tfrac{1}{2}x^2$ clearly $f$ is strictly concave but $f^*(y) = + \infty$ for all $y \in \Bbb R$. Which I don't think is considered as strictly convex function!

Mabey you need impose condition like boundedness!