On Extremal Finite Sets in the Sphere and Other Metric Spaces

Summary

The content of this paper is, thought slightly extended, based on my expository survey talk of the same title at the Montreal meeting: Algebraic Combinatorics and Extremal Set Theory, July 27 – August 2, 1986.

The aim of this paper is to study nice finite subsets in the sphere S^d and other (nice) metric spaces. This kind of study has a long history in mathematics. Its origin is perhaps traced back to the study of regular polyhedrons in R³ (by Platon?). In this paper, however, we restrict the scope of our discussion to the study of finite subsets which are extremal from the viewpoint of Delsarte theory (which we call Algebraic Combinatorics).

This paper consists of the following four sections:

1. §1. Harmonics on S^d and finite sets in the sphere S^d.

2. §2. Combinatorics of finite sets in compact symmetric spaces of rank one.

3. §3. Combinatorics of finite sets in noncompact symmetric spaces of rank one.

4. §4. Rigid t-designs in S^d.

In §1, we give a very brief and sketchy review of the theory of finite sets in S^d (i.e., spherical codes and designs) by Delsarte, Goethals and Seidel [18], which was the starting point of the study of finite sets in topological spaces from the view point of Algebraic Combinatorics.