Advertisement

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

Acute Angle Obtuse Angle Finite Linear Combination Parallel Hyperplane Convex Pentagon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations