  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.
