Journal of Mathematical Sciences

, Volume 131, Issue 1, pp 5406–5408 | Cite as

Estimating from Above the Perimeter of an Asymmetric Unit Disk in the Minkowski Plane

  • V. V. Makeev


It is proved that each convex planar figure K ⊂ ℝ2 contains a point O such that the perimeter of K computed with respect to the Minkowski distance function of the pair (K, O) is at most 9. If K is a triangle, this estimate is sharp. Bibliography: 7 titles.


Distance Function Unit Disk Asymmetric Unit Minkowski Plane Minkowski Distance 
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.


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  1. 1.
    H. Minkowski, Geometrie der Zahlen, Leipzig-Berlin (1910).Google Scholar
  2. 2.
    A. I. Shcherba, “On the estimate of the perimeter of the normalizing figure on the Minkowski plane,” in: 5th International Conference on Geometry and Topology in Memory of A. V. Pogorelov, Abstracts of Reports, Cherkassy (2003).Google Scholar
  3. 3.
    A. I. Shcherba, “On the estimate of the perimeter of the unit disk on the Minkowski plane,” in: Proc. Rubtsovsk Industry Institute, 12 (2003), pp. 96–107.Google Scholar
  4. 4.
    Yu. G. Reshetnyak, “An extremal problem in the theory of convex curves,” Uspekhi Mat. Nauk, 8, 125–126 (1953).Google Scholar
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    D. Laugwitz, “Konvexe Mittelpunktsbereiche und normierte Raume,” Math. Z., 61, 235–244 (1954).CrossRefGoogle Scholar
  6. 6.
    A. Besicovitch, “Measure of asymmetry of convex curves,” J. London Math. Soc., 23, 237–240 (1945).Google Scholar
  7. 7.
    V. V. Makeev, “On approximation of plane sections of a convex body,” Zap. Nauchn. Semin. POMI, 246, 174–183 (1997).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. V. Makeev
    • 1
  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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