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 Lipschitz 2nd-order derivatives?

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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.