Extremal problems of distance geometry related to energy integrals

Let K be a compact set, XT a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on K x K. We study the problem of determining for which , e Y1 (if any) the energy integral I(K, A) = IKIKf(X, y)dgXx)dgXy) 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 X is atomic we obtain good upper bounds for the sums of powers of all (2) 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 Szego for the calculation of transfinite diameters. 1. Background and summary of results. In this paper we will investigate a number of extremal problems in distance geometry. Our work is in many ways analogous to the study of energy integrals in classical potential theory. Let K be a compact set in a Euclidean space and 51 be a prescribed family of Borel measures (possibly signed) of total mass one supported by K. Suppose f is a continuous real-valued function on K x K. We consider the family of integrals having the form (1.1) I(K, = | f (x, y) d1(x)d1d(y), it E 1. A number of interesting questions naturally arise concerning 1(K), the supremum of the numbers I(K, pi) with it in Vl: (i) What is the numerical value of I(K)? (ii) Does there exist a y0 in V1I such that I(K, Io) = 1(K)? (iii) If y0 exists, is this measure unique? (iv) Can an extremal measure [t be explicitly produced? Received by the editors March 20, 1973 and, in revised form, April 20, 1973. AMS (MOS) subject classifications (1970). Primary 52A25, 52A40.

[1]  Roger A. Johnson.  Advanced Euclidean geometry (formerly titled: Modern geometry) : an elementary treatise on the geometry of the triangle and the circle , 1929 .

[2]  I. J. Schoenberg,et al.  Metric spaces and positive definite functions , 1938 .

[3]  R. Alexander,et al.  On the sum of distances betweenn points on a sphere , 1972 .

[4]  R. Alexander,et al.  On the sum of distances betweenn points on a sphere. II , 1972 .

[5]  E. Hille,et al.  Some geometric extremal problems , 1966, Journal of the Australian Mathematical Society.

[6]  I. J. Schoenberg.  On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space , 1937 .

[7]  Metrically homogeneous sets , 1968 .

[8]  I. J. Schoenberg.  Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[9]  E. Hille,et al.  Methods In Classical And Functional Analysis , 1972 .

[10]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[11]  P. Du Val,et al.  A Modern View of Geometry , 1962 .

[12]  H. Hasse.  Vorlesungen über Zahlentheorie , 1950 .

[13]  G. Pólya,et al.  Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. , 1931 .

[14]  L. Fejes Tóth,et al.  On the sum of distances determined by a pointset , 1956 .

[15]  H. Witsenhausen.  On the Maximum of the Sum of Squared Distances Under a Diameter Constraint , 1974 .

[16]  K. Stolarsky.  Sums of distances between points on a sphere. II , 1972 .

[17]  E. Netto,et al.  Lehrbuch der Combinatorik , 1902 .

[18]  Edmund Landau.  Vorlesungen über Zahlentheorie : aus der elementaren Zahlentheorie , 1950 .

[19]  Leonard M. Blumenthal,et al.  Theory and applications of distance geometry , 1954 .

[20]  L. Tóth,et al.  Über eine Punktverteilung auf der Kugel , 1959 .

[21]  E. Hille,et al.  Analytic Function Theory. Volume II , 1973 .

[22]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[23]  E. Hille.  Analytic Function Theory , 1961 .

[24]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[25]  G. Björck,et al.  Distributions of positive mass, which maximize a certain generalized energy integral , 1956 .

[26]  Metric inequalities and the zonoid problem , 1973 .

[27]  Günter Sperling.  Lösung einer elementargeometrischen Frage von FEJES TÒTH , 1960 .