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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
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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Keywords: Extremal problems of distance geometry, energy integrals, family of signed Borel measures, Euclidean <I>m</I>-sphere, great circle distance, ultraspherical harmonics, metric embedding, strictly metrically homogeneous, transfinite diameter, metric curvature
Article copyright: © Copyright 1974 American Mathematical Society