Exercise 6, § 5

Let $A$ be a closed set in $\mathbf{R}^n$ with the following properties; for every $x \in \mathbf{R}^n$, there exists one and only one point $y \in A$ such that $d(x, y) = d(x, A)$, where $d$ is the euclidean distance. Show that $A$ is convex. (Argue by reductio ad absurdum, considering a closed segment with end points $a, b$ in $A$ containing a point $c \in \mathscr{C} A$; there is a closed ball $B$ of centre $c$ contained in $\mathscr{C} A$; consider the set $\mathscr{B}$ of closed balls $S$ which contain $B$ and whose interiors do not meet $A$; show that the radii of these balls is bounded above, and deduce that there exists one of these balls $S_0$ whose radius $\rho$ is the largest possible. Then get a contradiction by proving that $S_0$ can only meet $A$ in a single point, and that this implies the existence in $\mathscr{B}$ of a ball of radius $> \rho$.)