An extremum property characterizing the n-dimensional regular cross-polytope

Abstract

In the spirit of the Genetics of the Regular Figures, by L. Fejes T\'oth, we prove the following theorem: If $2n$ points are selected in the $n$-dimensional Euclidean ball $B^n$ 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 for $2n$ points on the surface of the ball. We also generalize, in the same manner, a theorem of Davenport and Haj\'os on a set of $n+2$ points. As a corollary, we obtain a solution to the problem of packing $k$ unit $n$-dimensional balls $(n+2\le k\le 2n)$ into a spherical container of minimum radius.