Discrete & Computational Geometry

, Volume 62, Issue 4, pp 775–780 | Cite as

Acute Sets of Exponentially Optimal Size

  • Balázs GerencsérEmail author
  • Viktor Harangi


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.


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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