Sums of distances between points on a sphere. II

Stolarsky, Kenneth B.

doi:10.1090/S0002-9939-1973-0333995-9

Sums of distances between points on a sphere. II
HTML articles powered by AMS MathViewer

by Kenneth B. Stolarsky

Proc. Amer. Math. Soc. 41 (1973), 575-582

DOI: https://doi.org/10.1090/S0002-9939-1973-0333995-9

PDF | Request permission

Abstract:

Given $N$ points on a unit sphere in Euclidean $m$ space, $m \geqq 2$, 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 $m \geqq 5$ 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.

References

R. Alexander, On the sum of distances between $n$ points on a sphere, Acta Math. Acad. Sci. Hungar. 23 (1972), 443–448. MR 312395, DOI 10.1007/BF01896964
Ralph Alexander and Kenneth B. Stolarsky, Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc. 193 (1974), 1–31. MR 350629, DOI 10.1090/S0002-9947-1974-0350629-3
Göran Björck, Distributions of positive mass, which maximize a certain generalized energy integral, Ark. Mat. 3 (1956), 255–269. MR 78470, DOI 10.1007/BF02589412
L. Fejes Tóth, On the sum of distances determined by a pointset, Acta Math. Acad. Sci. Hungar. 7 (1956), 397–401 (English, with Russian summary). MR 107212, DOI 10.1007/BF02020534
Einar Hille, Some geometric extremal problems, J. Austral. Math. Soc. 6 (1966), 122–128. MR 0198349
John B. Kelly, Metric inequalities and symmetric differences, Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967) Academic Press, New York, 1970, pp. 193–212. MR 0264600

Om summen af afstandene mellem n punkter påen Kugleflade

13

Über den transfiniten Durchmesser

Kapazitätskonstante

von ebenen und räumlichen Punktmengen

165

Wolfgang M. Schmidt, Irregularities of distribution. IV, Invent. Math. 7 (1969), 55–82. MR 245532, DOI 10.1007/BF01418774
I. J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. of Math. (2) 38 (1937), no. 4, 787–793. MR 1503370, DOI 10.2307/1968835
Günter Sperling, Lösung einer elementargeometrischen Frage von Fejes Tóth, Arch. Math. 11 (1960), 69–71 (German). MR 112077, DOI 10.1007/BF01236910
Kenneth B. Stolarsky, Sums of distances between points on a sphere, Proc. Amer. Math. Soc. 35 (1972), 547–549. MR 303418, DOI 10.1090/S0002-9939-1972-0303418-3