SCHWEITZER 1982 Problem 7

Let $V$ be a bounded, closed, convex set in $\mathbb{R}^n$ , and denote by $r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $V$ ). Show that $r$ is the only real number with the following property: for any finite number of points in $V$ , there exists a point in $V$ such that the arithmetic mean of its distances from the other points is equal to $r$ .

Gy. Szekeres