Legendre transformation of a convex function and primal minimum

Question

I am having trouble in proving following property:

If $f$ is convex (and consequently $f^{**} = f$) and minimal in set $X$ exists, i.e. there is $x^* \in X$ such that $f^* = f(x^*) = \inf_{x \in X} f(x)$. Then it holds that $$f^* = \min_{x\in X} \max_{s\in dom~f^*} [<s,x> - f^*(s)] = \max_{s\in dom~f^*} \min_{x\in X} [<s,x> - f^*(s)]$$

Answer 1

I figured it should be proved as follows: $$f^*(0) = \sup_{x \in X} -f(x) = \inf_{x \in X} f(x) = f^*$$ $$\max_s \min_x [<s,x> - f^*(s)] \geq \min_x 0 - f^*(0) = f^*$$ and by min max > max min, it follows the equality as desired.