On designs in compact metric spaces and a universal bound on their size

Abstract:

For finite and compact infinite metric spaces, a concept of a (weighted) τ-design is introduced which depends on a choice of a substitution function. To estimate the minimum size of a τ-design a system of orthogonal polynomials is defined using the average measure of metric balls and the substitution function. A universal lower bound on the size of τ-designs is obtained with the help of the solution of the known extremum problem for systems of orthogonal polynomials. The concept of a τ-design and the bound considered coincide with those in the case of polynomial association schemes of Delsarte and the Euclidean sphere for the proper choices of the substitution functions. This bound is also calculated for some other spaces.