Sums of distances between points on a sphere. II

Abstract

Given N N points on a unit sphere in Euclidean m m space, m ≧ 2 m \geqq 2 , we show that the sum of all distances which they determine plus their discrepancy is a constant. As applications we obtain (i) an upper bound for the sum of the distances which for m ≧ 5 m \geqq 5 is smaller than any previously known and (ii) the existence of N N point distributions with small discrepancy. We make use of W. M. Schmidt’s work on the discrepancy of spherical caps.