Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by $$ f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}. $$ Then there exist well-known sufficient conditions such that $f^*$ is Lipschitz differentiable (e.g. $f$ is strongly convex). May I know whether there exist sufficient conditions such that $f^*$ is $C^2$ or even $C^2$ with local Lipschitz 2nd-order derivatives?
The motivation is that I would like to consider a function of the form: $$ f_\epsilon(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-I_C(x)-\epsilon g(x)\}, $$ where $C$ is a convex set (say a standard simplex). I was wondering under which condition on the convex function $g$ such that $f_\epsilon$ is a $C^2$-function.
For $n=1$, $C=[-1,1]$, we know $$ \sup_{x\in [-1,1]}\{x^Ty-\epsilon|x|^2/2\}= \begin{cases} x^2/(2\epsilon) &|x|\le \epsilon,\\ |x|-\epsilon/2 &|x|> \epsilon,\\ \end{cases} $$ which is not $C^2$. But if we choose $g(x)=1-\sqrt{1-x^2}$ for $x\in[-1,1]$, we can obtain a smooth function.
Similarly, for $n=2$, $C=\{a_1+a_2=1| a_1,a_2\ge 0\}$, we can choose $g=x_1\log(x_1)+x_2\log(x_2)$ to obtain the smooth entropy function.
