Extremal Problems of Distance Geometry Related to Energy Integrals

Let K be a compact set, <!-- MATH $\mathcal{M}$ --> a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on <!-- MATH $K \times K$ --> . We study the problem of determining for which <!-- MATH $\mu \in \mathcal{M}$ --> (if any) the energy integral <!-- MATH $I(K,\mu ) = \smallint_K {\smallint_K {f(x,y)d\mu (x)d\mu (y)} }$ --> 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 <!-- MATH $\mathcal{M}$ --> is atomic we obtain good upper bounds for the sums of powers of all 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.