2001/12/28 by Wlodzimierz Kuperberg, Kuperberg, Wlodzimierz
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #math.MG
paper · pdf · doi:10.48550/arxiv.math/0112290
4 pages
arxiv created 2001/12/28 · arxiv updated 2009/11/30
In the spirit of the Genetics of the Regular Figures, by L. Fejes Tóth, 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 for 2n points on the surface of the ball. We also generalize, in the same manner, a theorem of Davenport and Hajós 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.