Extremal problems of distance geometry related to energy integrals

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.