Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set

containing the origin in its interior, the support function of the set

\{y\in K^* \mid \exists x\in K\text{ such that } \langle x,y \rangle =1\} { y ∈ K ∗ ∣ ∃ x ∈ K  such that  ⟨ x , y ⟩ = 1 }

is the pointwise smallest among all sublinear functions

\sigma σ

such that

K=\{x \mid \sigma(x)\leq 1\} K = { x ∣ σ ( x ) ≤ 1 }