In my research, I've come across a problem that I haven't been able to find the answer to. I've boiled down my stumbling block into something that's fairly simply stated, but it doesn't quite fall solely into my field of study.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a convex differentiable 1D function and its Fenchel Dual (or Convex Conjugate) $f_*:\mathcal{Y}\rightarrow\mathbb{R}$ be defined on the closed set $\mathcal{Y}$ $$ f_*(y) = \sup_{x}\, \{xy-f(x)\}. $$
Then given a valid probability density $p(y)$ on $\mathcal{Y}$ and functional of the form
$$J[f]=\int\limits_{y\in\mathcal{Y}} f_*(y) p(y)dy,$$
how would one express $\frac{\delta J}{\delta f}$?
I'm not sure if I haven't found the answer because I don't know the right search terms, the class of problems hasn't attracted much study, or I've already found the answer but haven't recognized it as such. I'm also not sure if the problem is not possible in general without stricter conditions on $f$ or $f_*$ (e.g. strong convexity). Anyway, pointers would be much appreciated. Thanks!