Show that equality for support function of two compact convex sets implies that two sets are equal.

Question

Let $S\subseteq \mathbb{R}^n$.

The support function of set $S$ is defined as the following

$$ \sigma_S(x)=\sup_{y \in S} x^Ty $$ where $x \in \mathbb{R}^n$.

Let $F$ and $G$ be two compact convex sets in $\mathbb{R}^n$ such that $ \sigma_F(x)=\sigma_G(x)$.

Show that $F=G$.

Hint: use appropriate separation theorem.

Answer 1

This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $x\in G$. Hence $\sigma_G(y) \leq r<y^{T} u \leq \sigma_F(y)$ so $\sigma_F(y)\neq \sigma_G(y)$.