J Austral Math Soc Ser A 47 pp466--482, 1989.

On the Product of Distances to a Point Set on a Sphere

Gerold Wagner

Abstract

Let S be the surface of the unit sphere in three-dimensional euclidean space, and let w_N = (x₁, x₂, ..., x_N) be an N-tuple of points on S. We consider the product of mutual distances p(w_N) = Õ_{j ¹ k} |x_j - x_k| and, for the variable point x on S, the product of distances p(x, w_N) = Õ^N_j=1 |x - x_j| from x to the points of w_N. We obtain essentially best possible bounds for max_wN p(w_N) and for min_wN max_{x Î S} p(x, w_N).

1980 AMS Subject Classification (1985 Revision): 11K38, 52A40

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Authors

Gerold Wagner

Mathematisches Institut A, Pfaffenwaldring 57, D-7000 Stuttgart 80, Federal Republic of Germany.