On means of distances on the surface of a sphere. II (Upper bounds)

Abstract

Given TV points x1,…, xN on the unit sphere S in Euclidean d space (d ≥ 3), lower bounds for the deviation of the sum Σ |x-Xj|α, a > 1 - d; x ∈ S, from its mean value were established in terms of L1-norms in the first part of this paper. In the present part it is shown that these bounds are best possible. Our main tool is a multidimensional quadrature formula with equal weights. © 1992 by Pacific Journal of Mathematics.