Unit Distances and Diameters in Euclidean Spaces |
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Authors: | Konrad J. Swanepoel |
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Affiliation: | 1.Fakult?t für Mathematik,Technische Universit?t Chemnitz,Chemnitz,Germany;2.Department of Mathematical Sciences,University of South Africa,Pretoria,South Africa |
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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 and all n sufficiently large depending on d. This material is based upon work supported by the South African National Research Foundation. |
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Keywords: | Erdő s problem Combinatorial geometry Unit distance problem Lenz construction Diameters Erdő s– Simonovits stability theorem Erdő s– Stone theorem |
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