I have read following problem from the convex optimization book (by Stephen Boyd).
Although the solution is available but I want to prove it from another line of reasoning. We have to prove that if support functions of two convex closed sets are equal then the sets are equal. Suppose if for certain vector $y$, $x^*\in C$ and $z^* \in D$ are the maximizers in $C$ and $D$ respectively. Then we have to show that $x^*=z^*$ for every possible $y$. Since the support function is assumed to be equal for any $y$ therefore we can say that $x^*$ is not element wise greater than $z^*$ (and similarly $z^*$ is not element wise greater than $x^*$) since other wise we may have $y^T(x^*-z^*)\neq 0$ for $y$'s that have only positive elements. But I do not know how to show the other possibilities of $x^*\neq z^*$ are also not possible. I will be very thankful to you your help in this regard. Thanks in advance.