# Differentiability of the Moreau envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$$$e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(h),$$$$ where $$f$$ is a convex functional on a separable Hilbert space $$H$$.
Examples of results on differentiability are here, as well as Moreau's original papers. However, I didn't seem to come across any results about the twice differentiability of the operator... Are these results known/ what are the conditions on a convex function $$f$$ so that its Moreau envelope is twice continuously differentiable?

As you know, the Moreau enveloppe of a convex lower semicontinuous function is always Fréchet differentiable, with its gradient being $$1$$-lipschitz continuous. Actually, its gradient can be expressed as follows: $$\nabla e(f)(x) = x - \text{prox}_f (x)$$, where $$\text{prox}_f$$ is the proximal operator of $$f$$. So questioning the differentiability of $$\nabla e(f)$$ is equivalent to the one of $$prox_f$$.
One particular interesting example is $$f$$ being the indicator function of a closed convex set $$C$$ with nonempty interior, for which $$prox_f$$ becomes the projection $$\text{proj}_C$$. In this case, we know 1 that the projection is differentiable (almost) if and only if the set has a $$C^2$$ boundary.
For general functions $$f$$, we have that $$x^+ = \text{prox}_f(x) \iff x^+ = \text{proj}_{S_f(x^+)}(x)$$, where $$S_f(x^+)$$ is the sublevel set $$\{y \in H \ | \ f(y) \le f(x^+) \}$$. Even though $$S_f(x^+)$$ depends on $$x$$, this suggests that the projection onto sublevel sets should be differentiable. According to what precedes, we infer informally that the function should have lower sets of class $$C^2$$ (e.g. $$f$$ having a $$C^2$$ domain, and being twice differentiable in this domain). I unfortunately found no results on this question.
Terry Rockafellar has a paper on this topic in SIAM Optimization titled Generalized Hessian Properties of Regularized Nonsmooth Functions. See theorem 3.8. Rockafellar proves an equivalence between twice-differentiability of the Moreau envelope and twice epi-differentiability of the function $$f$$.