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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Extremal problems of distance geometry related to energy integrals


Authors: Ralph Alexander and Kenneth B. Stolarsky
Journal: Trans. Amer. Math. Soc. 193 (1974), 1-31
MSC: Primary 52A50
DOI: https://doi.org/10.1090/S0002-9947-1974-0350629-3
MathSciNet review: 0350629
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Abstract: Let K be a compact set, $ \mathcal{M}$ a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on $ K \times K$. We study the problem of determining for which $ \mu \in \mathcal{M}$ (if any) the energy integral $ I(K,\mu ) = \smallint_K {\smallint_K {f(x,y)d\mu (x)d\mu (y)} } $ is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when $ \mathcal{M}$ is atomic we obtain good upper bounds for the sums of powers of all $ (_2^n)$ distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Pólya and Szegö for the calculation of transfinite diameters.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0350629-3
Keywords: Extremal problems of distance geometry, energy integrals, family of signed Borel measures, Euclidean m-sphere, great circle distance, ultraspherical harmonics, metric embedding, strictly metrically homogeneous, transfinite diameter, metric curvature
Article copyright: © Copyright 1974 American Mathematical Society