Vol. 57, No. 2, 1975

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ISSN: 0030-8730
The sum of the distances to N points on a sphere

Kenneth Barry Stolarsky

Vol. 57 (1975), No. 2, 563–573
Abstract

How can the sum of λ-th powers (0 < λ < 2) of the Euclidean distances from the variable unit vector p to N fixed unit vectors p1,,pN be maximized or minimized? By means of an integral transform used in distance geometry, the problem can be reduced in certain cases to minimizing or maximizing sums of integer powers of the inner products (p,pi). In particular, a complete solution is obtained for the vertices of an m-dimensional octahedron.

Mathematical Subject Classification 2000
Primary: 10E99
Secondary: 52A40
Milestones
Received: 2 August 1974
Revised: 15 February 1975
Published: 1 April 1975
Authors
Kenneth Barry Stolarsky