Let $p:[0,\infty) \times \mathbb{R}^n \to \mathbb{R}$ be a smooth convex function for all $x \in \mathbb{R}^n$.
We define its Legendre transform (or convex conjugate) as
$ p^*(t,y)=\sup_{x \in \mathbb{R^n}}\left\{ x \cdot y - p(t,x)\right\} $
I know differentiability and convexity are dual concepts, so convexity of $p$ implies differentiability of $p^*$. However, I should be able to get smoothness of $p^*$ and Hessian $D^2_y p^*$ bounded away from zero and $+\infty$. How do I do this?
Thanks in advance for any helpful comments, ideas, etc.