Inside s-inner product sets and Euclidean designs

TL;DR

This paper investigates s-inner product sets in Euclidean space, establishing bounds on their size, proving non-existence in certain cases, and characterizing tight Euclidean designs as special inside s-inner product sets.

Contribution

It introduces the concept of inside s-inner product sets, provides bounds for their size, and characterizes tight Euclidean designs as those attaining these bounds.

Findings

Upper bounds for s-inner product sets on concentric spheres.

Non-existence of 2- or 3-inner product sets attaining the bound in certain cases.

Characterization of tight Euclidean designs as inside s-inner product sets.

Abstract

A finite set X in the Euclidean space is called an s-inner product set if the set of the usual inner products of any two distinct points in X has size s. First, we give a special upper bound for the cardinality of an s-inner product set on concentric spheres. The upper bound coincides with the known lower bound for the size of a Euclidean 2s-design. Secondly, we prove the non-existence of 2- or 3-inner product sets on two concentric spheres attaining the upper bound for any d>1. The efficient property needed to prove the upper bound for an s-inner product set gives the new concept, inside s-inner product sets. We characterize the most known tight Euclidean designs as inside s-inner product sets attaining the upper bound.