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On the angle sum of lines

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Abstract

What is the maximum of the sum of the pairwise (non-obtuse) angles formed by n lines in the Euclidean 3-space? This question was posed by Fejes Tóth in (Acta Math Acad Sci Hung 10:13–19, 1959). Fejes Tóth solved the problem for \({n \leq 6}\), and proved the asymptotic upper bound \({n^{2} \pi /5}\) as \({n \to \infty}\). He conjectured that the maximum is asymptotically equal to \({n^{2} \pi /6}\) as \({n \to \infty}\). The main result of this paper is an upper bound on the sum of the angles of n lines in the Euclidean 3-space that is asymptotically equal to \({3n^{2} \pi /16}\) as \({n \to \infty}\).

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Authors and Affiliations

  1. Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    F. Fodor, V. Vígh & T. Zarnócz

  2. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

    F. Fodor

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  1. F. Fodor
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  2. V. Vígh
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  3. T. Zarnócz
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Correspondence to V. Vígh.

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Fodor, F., Vígh, V. & Zarnócz, T. On the angle sum of lines. Arch. Math. 106, 91–100 (2016). https://doi.org/10.1007/s00013-015-0847-1

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