Abstract
A set of points in \({\mathbb {R}}^d\) is acute if any three points from this set form an acute triangle. In this note we construct an acute set in \({\mathbb {R}}^d\) of size at least \(1.618^d\). We also present a simple example of an acute set of size at least \(2^{{d}/{2}}\). Obtained bounds improve the previously best bound.
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References
Ackerman, E., Ben-Zwi, O.: On sets of points that determine only acute angles. Eur. J. Comb. 30(4), 908–910 (2009)
Danzer, L., Grünbaum, B.: Uber zwei Probleme bezüglich konvexer Körper von P. Erdős und von V.L. Klee. Math. Z. 79, 95–99 (1962)
Erdős, P., Füredi, Z.: The greatest angle among \(n\) points in the \(d\)-dimensional Euclidean space. In: Berge, C., et al. (eds.) Combinatorial Mathematics. Annals of Discrete Mathematics, vol. 17, pp. 275–283. North-Holland, Amsterdam (1983)
Harangi, V.: Acute sets in Euclidean spaces. SIAM J. Discrete Math. 25(3), 1212–1229 (2011)
Acknowledgements
We would like to thank Andrey Kupavskii and Alexandr Polyanskii for discussions that helped to improve the main result, as well as for their help in preparing this note. We would also like to thank Prof. Raigorodskii for introducing us to this problem and for his constant encouragement.
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Zakharov, D. Acute Sets. Discrete Comput Geom 61, 212–217 (2019). https://doi.org/10.1007/s00454-017-9947-y
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DOI: https://doi.org/10.1007/s00454-017-9947-y