A, B are convex, closed sets with not empty intersection.
$A^{\circ}$ is polar set of A.
Is it true:
$\overline{conv(A^{\circ} \cup B^{\circ})} = (A \cap B)^{\circ}?$
I was thinking that it is true and was trying to prove it as for case where sets contained 0 using property $(A \cup \{0\})^{\circ}= A^{\circ}$. I find hint in one article. It was something about circle segment without zero. I don't understand what to do with this fact.