How to prove this inequality in convex optimization?

Question

Suppose that $C$ is a proper convex cone, and $C^0$ is the dual cone. Then we have the following inequality $$\sup _{\|z\|_{2}=1, z \in C} h^{T} z \leq \inf _{z \in C^{\circ}}\|h-z\|_{2}.$$ I tried to prove this by writing out the dual of the original problem, but ran into some issues.

Answer 1

Let $z \in C$ with $\|z\|_2 = 1$ and $v \in C^\circ$ be given. Then, $$ h^\top z \le (h - v)^\top z \le \|h-v\|_2.$$

This implies your inequality.