AN EXTREMUM PROPERTY CHARACTERIZING THE n-DIMENSIONAL REGULAR CROSS-POLYTOPE

In the spirit of the Genetics of the Regular Figures, by L. Fejes Tóth [FT, Part 2], we prove the following theorem: If 2n points are selected in the n-dimensional Euclidean ball Bn so that the smallest distance between any two of them is as large as possible, then the points are the vertices of an inscribed regular cross-polytope. This generalizes a result of R.A. Rankin [R] for 2n points on the surface of the ball. We also generalize, in the same manner, a theorem of Davenport and Hajós [DH] on a set of n+ 2 points. As a corollary, we obtain a solution to the problem of packing k unit n-dimensional balls (n + 2 ≤ k ≤ 2n) into a spherical container of minimum radius.