Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set ${y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1}$ is the pointwise smallest among all sublinear functions $\sigma$ such that $K=\{x:\sigma (x)\le 1\}$.