Let $C$ be a non-empty closed convex subset of $\mathbb{R}^d$ and let $\sigma_C : \mathbb{R}^d \rightarrow \mathbb{R} \cup \{+\infty\}$ be defined by:
$$\sigma_C(p) := \sup\{p \cdot x, x \in C\}$$
show that
$$C = \left\{x \in \mathbb{R}^d : p \cdot x \leqslant \sigma_C(p), \forall p \in \mathbb{R}^d\right\}$$
(so that $C$ is an intersection of closed half-spaces).