On means of distances on the surface of a sphere (lower bounds)

Gerold Wagner

Vol. 144 (1990), No. 2, 389–398

DOI: 10.2140/pjm.1990.144.389

Abstract

Given N points x₁,x₂,…,x_N on a unit sphere S in Euclidean d space (d ≥ 3), we investigate the α-sum ∑ |x−x_j|^α, α > 1 −d, of their distances from a variable point x on S. We obtain an essentially best possible lower bound for the L¹-norm of its deviation from the mean value. As an application, we prove similar bounds for the α-sums ∑ |x_j − x_k|^α of mutual distances.

Mathematical Subject Classification 2000

Primary: 52A40

Milestones

Received: 23 September 1988

Published: 1 August 1990