Let $C$ and $D$ be two convex sets. And suppose $C\cap D\neq \emptyset$. Let $x^*$ is the solution to the optimization problem:

$$\min_{x\in C} \max_{y \in D} |x-y|^2$$

Is it true that $x^* \in D$. Without the constraint $x\in C$ this is the minimum enclosing ball problem and the statement is true. I am almost positive this should be true too but can't seem to find a proof.