An extremal characterization of regular simplices via the addition theorem for Gegenbauer polynomials

Abstract

Let p~, ..., Pn+2 be unit vectors in E n+~. Then the sum of the squares of all the (N+2, 2) inner products which they determine is minimal if and only if + signs can be chosen so that the _+p~ are the vertices of a regular simplex. The proof involves a special case of the addition theorem for Gegenbauer polynomials.* AMS(MOS) subject classifications (1970). Primary 52A25, 52A40.Geometriae Dedicata 5 (1976) 229-238. All Rights Reserved