My question comes from the famous book: convex analysis, Rockafellar Ch. 13 p.113
Note:
- $\delta^*(x^* \mid C) = \operatorname{sup}\{\langle x,x^*\rangle \mid x\in C\}$: a support function of a convex set $C$.
My question is how to explain the part underscored by red line? Why it is the case? I am also confused about the case where ri$(C)$ = int$(C)$ and the relation between hyperplane.
I know the concept of relative interior; however, I still cannot understand what the author was trying to say.