Discrete & Computational Geometry

, Volume 49, Issue 3, pp 589–600 | Cite as

The Small Octagons of Maximal Width

  • Charles Audet
  • Pierre Hansen
  • Frédéric Messine
  • Jordan Ninin
Article

Abstract

The paper answers an open problem introduced by Bezdek and Fodor (Arch. Math. 74:75–80, 2000). The width of any unit-diameter octagon is shown to be less than or equal to \(\frac{1}{4}\sqrt{10 + 2\sqrt{7}}\) and there are infinitely many small octagons having this optimal width. The proof combines geometric and analytical reasoning as well as the use of a recent version of the deterministic and reliable global optimization code IBBA based on interval and affine arithmetics. The code guarantees a certified numerical accuracy of \(1\times 10^{-7}\).

Keywords

Polygon Octagon Width Diameter 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Charles Audet
    • 1
  • Pierre Hansen
    • 2
  • Frédéric Messine
    • 3
  • Jordan Ninin
    • 4
  1. 1.GERAD and Département de Mathématiques et de Génie IndustrielÉcole Polytechnique de MontréalMontrealCanada
  2. 2.GERAD and Département des Méthodes QuantitativesHEC MontréalMontrealCanada
  3. 3.ENSEEIHT-IRITUniversity of ToulouseToulouse Cedex 7France
  4. 4.IHSEV Team, LAB-STICC, UMR-CNRS 3162ENSTA-BretagneBrestFrance

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