Jung's theorem

Remarks

Take the smallest closed ball containing $S$; by a compactness argument its centre lies in the convex hull of the points of the closure of $S$ on its boundary sphere. Since this centre is a convex combination of at most $n + 1$ such points, a computation with these points and the inequality $\abs{x_i - x_j} \le d$ gives the bound. A classical alternative uses Helly's theorem to reduce to sets of $n + 1$ points. The bound is sharp: the vertices of a regular $n$-simplex of edge $d$ have diameter $d$ and circumradius exactly $d \sqrt{n / (2(n+1))}$. Example: for $n = 2$ the radius is $d / \sqrt{3}$, the circumradius of an equilateral triangle of side $d$. For $n = 1$ the radius is $d/2$, as expected for a segment.

Going deeper

For a student

The trivial bound is $R \le d$: a ball of radius $d$ centred at any point of $S$ contains $S$. Jung's theorem gives the optimal constant, which grows from $1/2$ for $n = 1$ towards $1/\sqrt{2}$ as $n \to \infty$. It is a typical result of convex geometry that relates two measures of size, the diameter and the circumradius. It is often used together with the isodiametric inequality and in Borsuk-type partition problems.