Proof of uniqueness of convex sets using their support functions

Question

For a non-empty set $C \in \mathbb{R}^n$, the support function is $S_c(y)=sup_{x\in C}y^Tx$. We have two closed convex sets $A$ and $B$.

Using the fact that the support function is an extended-real valued convex function, I am not sure how to prove that the two sets are equal (that is, $A=B$) if and only if their support functions are equal.

Answer 1

Apparently if $A = B$, $\forall y: S_A(y)=S_B(y)$.

If $A \neq B$, without loss of generality, assume $\exists x \in A: x \notin B$. Since $B$ is closed, $x$ and $B$ could be strictly separated with a hyperplane [cvxbook Example 2.20]. That is to say $$ \exists y: \forall z \in B, y^Tx > y^Tz $$ Therefore $S_B(y) < y^Tx \leq S_A(y)$, so $S_A$ and $S_B$ are not equal at $y$.