Extremal Problems of Distance Geometry Related to Energy Integrals

Abstract

Let K be a compact set, % a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and / a continuous real-valued function on Kx K. We study the problem of determining for which /x e% (if any) the energy integral 1(K, fi) = Siefi(ßx' y)df4.x)df/.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 % 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 Polya and Szegö for the calculation of transfinite diameters.

Distance geometry in quasihypermetric spaces. III

“…In Section 3 of [19], for example, we discussed briefly some connections between the quasihypermetric property and L 1 -embeddability and between the quasihypermetric property and the metric embedding ideas of Schoenberg [22]. Also, embedding arguments based around and extending Schoenberg's ideas were used in [2] by Alexander and Stolarsky to obtain information on M (X) when X is a subset of euclidean space, and in [4] by Assouad to characterize the hypermetric property in finite metric spaces.…”

“…We note that Alexander and Stolarsky [2,Theorem 3.3] give a simple algorithm involving the solution of a system of linear equations for the computation of M (X) when X is a (strictly quasihypermetric) finite subset of euclidean space. …”

On the sum of distances betweenn points on a sphere. II

“…However, from the viewpoint of integral geometry (9) is a special case of (6), which holds for a broad class of metrics and arbitrary compact sets in Euclidean space as discussed in [2]. Thus the problems studied in [3] and [5] continue to be a source of interesting results.…”

An extremal characterization of regular simplices via the addition theorem for Gegenbauer polynomials

“…In this section we shall follow the notation of [3]; this is essentially the notation of Chapters X and XI of [7]. Note that on p. 21, line 8, of [3] In fact (see [3], for example) all the ck are positive and there are explicit formulas for the hk(P) in terms of a fixed basis of orthonormal vectors el, e2, ..., eN+ 1.…”

“…In this section we shall follow the notation of [3]; this is essentially the notation of Chapters X and XI of [7]. Note that on p. 21, line 8, of [3] In fact (see [3], for example) all the ck are positive and there are explicit formulas for the hk(P) in terms of a fixed basis of orthonormal vectors el, e2, ..., eN+ 1. (These formulas are usually written in terms of the spherical co-ordinates of p, but the spherical co-ordinate system can be transformed back into a rectangular co-ordinate system with the basis el ..... eN+l '.)…”