Support-function subgradients as exposed faces ($\partial {\sigma }_C(x)=\{p\in C:⟨p,x⟩={\sigma }_C(x)\}$)

For a nonempty closed convex set $C$, the finite-valued support function has subgradients exactly at the maximizing points of $C$. This follows from Fenchel–Young inequality and the conjugacy between the support function and the indicator functional of a constraint set. For compact $C$, the exposed maximizing set is nonempty.