Unit distances and diameters in Euclidean spaces

TL;DR

This paper characterizes the maximum number of unit distances and diameters in large point sets in Euclidean spaces of dimension four and higher, identifying specific constructions that achieve these maxima and determining exact counts for certain dimensions.

Contribution

It establishes that maximum configurations are specific Lenz constructions for all dimensions d >= 4 and large n, and provides exact counts for even dimensions d >= 6.

Findings

Maximum number of unit distances in high-dimensional Euclidean spaces is achieved by Lenz constructions.

Exact maximum counts for unit distances in even dimensions d >= 6.

Exact maximum counts for diameters in dimensions d >= 4.

Abstract

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit distances for all even d >= 6, and the exact maximum number of diameters for all d >= 4, for all $n$ sufficiently large, depending on d.