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.
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13 April 2018
Due to a typesetting error some mistakes were introduced; most importantly at the end the third page of the article in the displayed equation
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Editor in Charge: János Pach
The first author was supported by NKFIH (National Research, Development and Innovation Office) Grant PD 121107. The second author was supported by “MTA Rényi Lendület Véletlen Spektrum Kutatócsoport”.
The original article was revised to correct errors introduced during typesetting.
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Gerencsér, B., Harangi, V. Acute Sets of Exponentially Optimal Size. Discrete Comput Geom 62, 775–780 (2019). https://doi.org/10.1007/s00454-018-9985-0
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DOI: https://doi.org/10.1007/s00454-018-9985-0