Journal of Geometry

, Volume 106, Issue 1, pp 75–83 | Cite as

A remark on perimeter–diameter and perimeter–circumradius inequalities under lattice constraints

  • Bernardo González Merino
  • Matthias Henze


In this note, we are concerned with planar convex sets whose interior is free from points of the integer lattice. Such lattice-free convex sets have been extensively studied with respect to determining sharp relations between any two of the geometric functionals: perimeter, diameter, circumradius, inradius, minimal width and area. The cases perimeter–diameter and perimeter–circumradius have been left open and are the subject of our investigations. We formulate precise conjectures on linear inequalities for these settings and prove them in special yet illustrative cases. Moreover, we obtain sharp non-linear inequalities that hold unconditionally.

Mathematics Subject Classification (2010)

Primary 52A10 Secondary 52A40 52C05 


Lattice-free convex set perimeter diameter circumradius 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Awyong P.W., Scott P.R.: Inradius and circumradius for planar convex bodies containing no lattice points. Bull. Aust. Math. Soc. 59(1), 163–168 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Awyong P.W., Scott P.R.: New inequalities for planar convex sets with lattice point constraints. Bull. Aust. Math. Soc. 54(3), 391–396 (1996)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Barvinok, A.: Integer Points in Polyhedra, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)Google Scholar
  4. 4.
    Bonnesen, T., Fenchel, W.: Theory of Convex Bodies, BCS Associates, Moscow (1987) (Translated from the German and edited by L. Boron, C. Christenson and B. Smith)Google Scholar
  5. 5.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry, Problem Books in Mathematics. Springer, New York (1994) (Corrected reprint of the 1991 original, Unsolved Problems in Intuitive Mathematics, II)Google Scholar
  6. 6.
    Gritzmann, P., Wills, J.M.: Lattice Points, Handbook of Convex Geometry, vols. A, B, pp. 765–797. North-Holland, Amsterdam (1993)Google Scholar
  7. 7.
    Gruber, P.M.: Convex and Discrete Geometry Springer, Berlin (2007)Google Scholar
  8. 8.
    Henk M., Tsintsifas G.A.: Some inequalities for planar convex figures. Elem. Math. 49(3), 120–125 (1994)MATHMathSciNetGoogle Scholar
  9. 9.
    Hernández Cifre, M.A., Scott, P.R., An isodiametric problem with lattice-point constraints. Bull. Aust. Math. Soc. 57(2), 289–294 (1998)Google Scholar
  10. 10.
    Hernández Cifre, M.A., Segura Gomis, S.: Some inequalities for planar convex sets containing one lattice point. Bull. Aust. Math. Soc. 58, 159–166 (1998)Google Scholar
  11. 11.
    Hillock, P.W., Scott, P.R.: Inequalities for lattice constrained planar convex sets. JIPAM. J. Inequal. Pure Appl. Math. 3(2), Article 23 (2002) (electronic)Google Scholar
  12. 12.
    Kubota T.: Eine Ungleichheit für die Eilinien. Math. Z. 20(1), 264–266 (1924)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Minkowski, H.: Geometrie der Zahlen. Teubner, Leipzig (1910)Google Scholar
  14. 14.
    Scott, P.R.: Area-diameter relations for two-dimensional lattices. Math. Mag. 47, 218–221 (1974)Google Scholar
  15. 15.
    Scott, P.R.: Two inequalities for convex sets in the plane. Bull. Aust. Math. Soc. 19(1), 131–133 (1978)Google Scholar
  16. 16.
    Scott P.R.: Modifying Minkowski’s theorem. J. Number Theory 29(1), 13–20 (1988)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaMurciaSpain
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany

Personalised recommendations