The problem:
Assume $A$ is a bounded subset of a Hilbert space $H$. Let $r$ be the infimum of the radii of closed balls containing $A$, so
$r = \inf \{s \geq 0 $ $\vert$ there exists $x \in H$ such that $\Vert y - x \Vert \leq s$ for all $y \in A \}$
Show that there exists a unique closed ball of radius $r$ containing $A$. The center of this ball is called the circumcenter of $A$.
Show that if $A$ is convex and closed, then the circumcenter of $A$ lies in $A$.
I don't even know where to begin with this problem. Any help is greatly appreciated!