Unit Distances and Diameters in Euclidean Spaces

摘要

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 and all n sufficiently large depending on d.