Skip to main content
SpringerLink
Log in

Large triangles contained in the unit disk of a Minkowski plane

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are \({\frac{3}{2}}\) . We also prove that C permits to inscribe a triangle whose sides are of lengths at least \({\frac{3}{2}}\) in the positive orientation, or that they are of lengths at least \({\frac{3}{2}}\) in the negative orientation. The ratio \({\frac{3}{2}}\) in both the theorems is best possible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bezdek, A., Fodor, F., Talata, I.: Applications of inscribed affine regular polygons in convex disks. In: Proceedings of International Scientific Conference on Mathematics, Žilina, pp. 19–27 (1998)

  2. Chakerian G.D., Talley W.K.: Some properties of the self-circumference of convex sets. Arch. Math. 20, 431–443 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  3. Doyle P.G., Lagarias J.C., Randall D.: Self-packing of centrally symmetric convex bodies in R 2. Discrete Comput. Geom. 8, 171–189 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fabińska E., Lassak M.: Large equilateral triangles in the unit disk of Minkowski plane. Beitr. Alg. Geom. 24, 517–525 (2004)

    Google Scholar 

  5. Goła̧b S.: Some metric problems of the geometry of Minkowski. Trav. Acad. Mines Cracovie 6, 1–79 (1932)

    Google Scholar 

  6. Goła̧b S., Härlen H.: Minkowskische Geometrie. Monatsh. für Math. Physik 38, 387–398 (1931)

    Article  Google Scholar 

  7. Grünbaum B.: Self-circumference of convex sets. Coll. Math. 13, 55–57 (1964)

    MATH  Google Scholar 

  8. Lassak M.: Affine-regular hexagons of extreme areas inscribed in a centrally symmetric convex body. Adv. Geom. 3, 45–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Martini H., Swanepoel K.J., Weiss G.: The geometry of Minkowski spaces—a survey. Part 1. Expo. Math. 19, 97–142 (2001)

    MATH  MathSciNet  Google Scholar 

  10. Minkowski, H.: Diophantische Approximationen. Teubner, Leipzig (1907). Reprinted: Physika, Würzburg (1961)

Download references

Author information

Authors and Affiliations

  1. University of Technology, Kaliskiego 7, 85-796, Bydgoszcz, Poland

    Ewa Fabińska & Marek Lassak

  2. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warsaw, Poland

    Marek Lassak

Authors
  1. Ewa Fabińska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marek Lassak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marek Lassak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fabińska, E., Lassak, M. Large triangles contained in the unit disk of a Minkowski plane. J. Geom. 95, 31–39 (2009). https://doi.org/10.1007/s00022-009-0019-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00022-009-0019-1

Mathematics Subject Classification (2000)

Keywords

Search

Navigation