Unit distances and diameters in Euclidean spaces

Details Unit distances and diameters in Euclidean spaces Unit distances and diameters in Euclidean spaces

2007-07-02

arxiv.org/abs/0707.0213v1

Discrete and Computational Geometry 49 (2009), 1--27.

Mathematics

Metric Geometry

Scientific paper

10.1007/s00454-008-9082-x

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.

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