Inside s-inner product sets and Euclidean designs

2009/08/26 by Hiroshi Nozaki, Nozaki, Hiroshi
Engineering · Mathematics · #05B30 #52C99 #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Optimization and Packing Problems #graph theory and CDMA systems #math.CO #msc:05B30 #msc:52C99

paper · pdf · doi:10.48550/arxiv.0908.3801

9 pages, no figure

openalex publication_date 2009/08/26 · arxiv created 2011/04/19 · arxiv updated 2011/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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