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On a question of Makai and Martini on bodies of constant width

  • Alexandr Polyanskii
Original Paper
  • 82 Downloads

Abstract

We aim to answer the following question from Makai Jr and Martini (1990): Question Are the following properties equivalent for a convex body \(B\) in \({\mathbb R}^n\) with diameter \(1\):

1. Any \(n\) mutually perpendicular non-degenerate chords, having a common point, have total length at least \(1\).

2. \(B\) is a body of constant width \(1\).

Keywords

Convex geometry Bodies of constant width Discrete geometry 

Mathematics Subject Classification

52A20 

Notes

Acknowledgments

The author expresses his thanks to A. V. Akopjan and R. N. Karasev for turning his attention to this problem.

References

  1. Alonso, J., Martini, H., Mustafaev, Z.: On orthogonal chords in normed planes. Rocky Mountain J. Math. 41, 23–35 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper, Ergebn. d. Math. u. ihrer Grenzgeb. Springer, Berlin (1934)CrossRefGoogle Scholar
  3. Makai Jr, E., Martini, H.: A new characterization of convex plates of constant width. Geometriae Dedicata 34, 199–209 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2014

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyMoscowRussia

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