Every large point set has an obtuse angle

  • Martin Aigner
  • Günter M. Ziegler

Abstract

Around 1950 Paul Erdös conjectured that every set of more than 2 d points in ℝ d determines at least one obtuse angle,that is, an angle that is strictly greater than \(\frac{\pi }{2}\). In other words, any set of points in ℝ d which only has acute angles (including right angles) has size at most 2 d . This problem was posed as a “prize question” by the Dutch Mathematical Society — but solutions were received only for d = 2 and for d = 3.

Keywords

Hull 

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References

  1. [1]
    L. Danzer & B. Grünbaum: Über zwei Probleme bezüglich konvexer Körper von P Erdös und von V L. Klee, Math. Zeitschrift 79 (1962), 95–99.MATHCrossRefGoogle Scholar
  2. [2]
    P. Erdős & Z. Füredi: The greatest angle among n points in the d-dimensional Euclidean space, Annals of Discrete Math. 17 (1983), 275–283.Google Scholar
  3. [3]
    H. Minkowski: Dichteste gitterförmige Lagerung kongruenter Körper, Nachrichten Ges. Wiss. Göttingen, Math.-Phys. Klasse 1904, 311–355.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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