Jung’s theorem is, in its simplest form, an inequality relating the diameter \(D\) and circumradius \(R\) of compact sets in \(\mathbb{R}^d\) by giving a lower bound for \(D/R\) which depends only on \(d\). The author has already extended this theorem in a previous paper to include additional inequalities involving the edge-lengths of inscribed simplexes.
In this paper, the theorem is further extended to spherical and hyperbolic spaces. As these spaces have a natural length scale, the corresponding inequalities are a little more complicated, involving trigonometric and hyperbolic functions respectively.
The author also investigates precisely which pairs \((R,D)\) are realizable as the circumradius and diameter respectively of a compact set in \(R^d\), \(H^d\), of \(S^d\). In the Euclidean and hyperbolic cases, and the “small-cap” spherical case with circumradius less than \(\pi/2\), the answers are unsurprising and given by the inequalities derived earlier.
However, in the “large-cap” spherical case, for spherical sets that do not fit into a single hemisphere, combinatorial considerations arise as well. It is clear that \(D \geq R\), and the bound \(D = R = \pi(1- 1/(2k+1))\) is actually attained by \(2k+1\) points equally spaced around \(S^1\) (and by a related construction in \(S^d\)). While the author does not make this explicit, it is easily seen (in the case \(d = 1\)) that for intermediate values of \(R\), \(D = R\) is not attained; so that the boundary of the set of realizable points must have infinitely many scallops in it. The author traps this boundary between the line \(D = R\) and a zigzag polygonal arc, whose vertices are alternately on and above \(D = R\).