Support-function subgradients as exposed faces
= Support-function subgradients as exposed faces
{title2=$\partial\sigma_C(x)=\{p\in C:\langle p,x\rangle=\sigma_C(x)\}$}
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.