论文信息 - Sums of distances between points on a sphere. II

Sums of distances between points on a sphere. II

ABSTRACr. 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.

K. Stolarsky

[1] R. Alexander,et al. On the sum of distances betweenn points on a sphere. II , 1972 .

[2] Wolfgang M. Schmidt. Irregularities of distribution. IV , 1969 .

[3] W. Schmidt. Irregularities of distribution , 1968 .

[4] E. Hille,et al. Some geometric extremal problems , 1966, Journal of the Australian Mathematical Society.

[5] Günter Sperling. Lösung einer elementargeometrischen Frage von FEJES TÒTH , 1960 .

[6] L. Fejes Tóth,et al. On the sum of distances determined by a pointset , 1956 .

[7] G. Björck,et al. Distributions of positive mass, which maximize a certain generalized energy integral , 1956 .

[8] I. J. Schoenberg,et al. Metric spaces and positive definite functions , 1938 .

[9] I. J. Schoenberg. On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space , 1937 .

[10] G. Pólya,et al. Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. , 1931 .