Suppose that $f\in \Gamma_0(H)$, that is $f$ is a lsc, convex, and proper functional from a Hilbert space $H$ to the base-field $\mathbb{R}$.
Is it possible to reconstruct $f$, from the Fenchel-Moreau subdifferential $$ \begin{aligned} &\partial f: dom(H) \rightrightarrows H\\ & \partial f: x\mapsto \{ y \in H :(\forall z \in H)\, \langle y,\, z-x\rangle \leq f(z)-f(x) \}? \end{aligned} $$
Intuition
I expect (more or less) since if we take the special case where
- $H=\mathbb{R}^d$ and $f$ differentiable (ie: $\partial f = \{\nabla f\}$),
- there is some $p \in \mathbb{R}^d$ such that $f(y)=0$,
then the Fundamental theorem of calculus for line integrals implies that
$$
f(x)= \int_{\gamma_{[x:y]}} \nabla f(s) \cdot ds,
$$
where $\gamma_{[x:y]}$ is any smooth curve joining $x$ to $y$.
Therefore, in this case, the subdifferential may be used to reconstruct the function $f$ perfectly.