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Archiv der Mathematik

, Volume 106, Issue 1, pp 91–100 | Cite as

On the angle sum of lines

  • F. Fodor
  • V. Vígh
  • T. Zarnócz
Article
  • 76 Downloads

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 \({n \to \infty}\).

Keywords

Angle sum of lines Upper bound 

Mathematics Subject Classification

52C35 

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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of Geometry, Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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