I would like you to give me some advice on how I can use the advice that this problem gives me, I managed to solve the problem using only properties but I did not use the suggestion.
Let $C, D \subset \mathbb{R}^{n}$ are non-empty and convex, such that $0 \notin \textbf{ri} (C-D)$ wanted to prove that: $$\inf_{x\in C}\left \langle x,z \right \rangle\geq \sup_{y\in D}\left \langle y,z \right \rangle$$
Suggestion: proof that $\textbf{cone}(C-D)\neq \mathbb{R}^{n}$
My steps:Since $0 \notin \textbf{ri} (C-D)$ by proper separation theorem, it follows that there exist a hiperplane proporty separating $C-D$ and the origin, then we will have the inequality we require, but then I don't use the problem suggestion, any idea how I can use it?
