On the volume product of polygons

Article

Abstract

We present a method that allows us to prove that the volume product of polygons in ℝ2 with at most n vertices is bounded from above by the volume product of regular polygons with n vertices. The same method shows that the volume product of polygons is bounded from below by the volume product of triangles (or parallelograms in the centrally symmetric case). These last results give a new proof of theorems of K. Mahler. The cases of equality are completely described.

Keywords

Convex bodies Volume-product Polygons Affinely-regular 

Mathematics Subject Classification (2010)

52A20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blaschke, W.: Über affine geometrie VII: Neue Extremeigenschaften von Ellipse und Ellipsoid. Ber. Verh. Sächs. Akad. Wiss. Leipz. Math.-Phys., 69, 306–318 (1917) Google Scholar
  2. 2.
    Böröczky, K., Jr., Makai, E., Jr., Meyer, M., Reisner, S.: Volume product in the plane—lower estimates with stability. Preprint (2010) Google Scholar
  3. 3.
    Coxeter, H.S.M., Affinely regular polygons. Abh. Math. Semin. Univ. Hamb. 34, 38–58 (1969) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Fisher, J.C., Jamison, R.J.: Properties of affinely regular polygons. Geom. Dedic. 69, 241–259 (1998) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Kim, J., Reisner, S.: Local minimality of the volume-product at the simplex. Mathematika 57, 121–134 (2011) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lutwak, E.: On the Blaschke-Santaló Inequality, Discrete Geometry and Convexity (New York, 1982). Ann. New York Acad. Sci., vol. 440, pp. 106–112. N.Y. Acad. Sci., New York (1985) Google Scholar
  7. 7.
    Mahler, K.: Ein Minimalproblem für konvexe Polygone. Math. (Zutphen) B 7, 118–127 (1939) MathSciNetGoogle Scholar
  8. 8.
    Mahler, K.: Ein Übertragungsprinzip für konvexe Körper. Časopis Pěst. Mat. Fys. 68, 93–102 (1939) MathSciNetGoogle Scholar
  9. 9.
    Meyer, M.: Convex bodies with minimal volume product in ℝ2. Monatshefte Math. 112, 297–301 (1991) MATHCrossRefGoogle Scholar
  10. 10.
    Meyer, M., Pajor, A.: On the Blaschke-Santaló inequality. Arch. Math. 55, 82–93 (1990) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Petty, C.M.: Affine isoperimetric problems. Discrete geometry and convexity (New York, 1982). Ann. New York Acad. Sci., vol. 440, pp. 113–127. N.Y. Acad. Sci., New York (1985) Google Scholar
  12. 12.
    Reisner, S.: Zonoids with minimal volume-product. Math. Z. 192, 339–346 (1986) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Saint Raymond, J.: Sur le volume des corps convexes symétriques. Séminaire d’Initiation à l’Analyse, 1980–1981, Université Paris VI (1981) Google Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2011

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050)Université Paris-Est Marne-la-ValléeMarne-la-Vallée cedex 2France
  2. 2.Department of MathematicsUniversity of HaifaHaifaIsrael

Personalised recommendations