Sums of Distances Between Points on a Sphere. II

Given N points on a unit sphere in Euclidean m space, m 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 is smaller than any previously known and (ii) the existence of N point distributions with small discrepancy. We make use of W. M. Schmidt’s work on the discrepancy of spherical caps.