Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

Question

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in \dot B$. \begin{equation} \lambda (u+x)+(1-\lambda (v+y))\in \dot(A+B) ...(interior) \end{equation} How would we show this for a concave function such as the unit circle?