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Acute Sets of Exponentially Optimal Size

  • Balázs Gerencsér
  • Viktor Harangi
Article
  • 54 Downloads

Abstract

We present a simple construction of an acute set of size \(2^{d-1}+1\) in \(\mathbb {R}^d\) for any dimension d. That is, we explicitly give \(2^{d-1}+1\) points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than \(2^d\). Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order \(\varphi ^d\) where \(\varphi = (1+\sqrt{5})/2 \approx 1.618\) is the golden ratio.

Keywords

Acute set Acute angles Hypercube Strictly antipodal 

Mathematics Subject Classification

51M04 51M15 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MTA Alfréd Rényi Institute of MathematicsBudapestHungary
  2. 2.Department of Probability and StatisticsEötvös Loránd UniversityBudapestHungary

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