Let $S$ be some (not necessarily convex) subset of positive semidefinite matrices.
What is the convex conjugate of the function $$ f(x) = \sup_M \left\{ x^T M x \;:\; M \in S \right\},$$ that is, what is $f^*(y) = \sup_x \{ x^T y - f(x) \}$?
I'm struggling with this one because there are essentially two maximization problems involved, and I don't really have any idea about how to approach this. I'd appreciate any help.