Acta Mathematica Hungarica

, Volume 154, Issue 2, pp 457–469 | Cite as

Improved bounds on the diameter of lattice polytopes

  • A. DezaEmail author
  • L. Pournin


We show that the largest possible diameter \({\delta(d,k)}\) of a d-dimensional polytope whose vertices have integer coordinates ranging between 0 and k is at most \({kd - \lceil2d/3\rceil-(k-3)}\) when \({k\geq3}\) . In addition, we show that \({\delta(4,3)=8}\) . This substantiates the conjecture whereby \({\delta(d,k)}\) is at most \({\lfloor(k+1)d/2\rfloor}\) and is achieved by a Minkowski sum of lattice vectors.

Mathematics Subject Classification

primary 52B20 90C27 secondary 52C45 90C05 

Key words and phrases

lattice polytope diameter Minkowski sum 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acketa D., Žunić J.: On the maximal number of edges of convex digital polygons included into an m × m-grid. J. Combin. Theory A, 69, 358–368 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    X. Allamigeon, P. Benchimol, S. Gaubert and M. Joswig, Long and winding central paths, arXiv:1405.4161 (2014).
  3. 3.
    A. Balog and I. Bárány, On the convex hull of the integer points in a disc, in: Proceedings of the Seventh Annual Symposium on Computational Geometry (1991), pp. 162–165.Google Scholar
  4. 4.
    Bonifas N., Di Summa M., Eisenbrand F., Hähnle N., Niemeier M.: On sub-determinants and the diameter of polyhedra. Discrete Comput Geom., 52, 102–115 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Borgwardt, J. De Loera and E. Finhold, The diameters of transportation polytopes satisfy the Hirsch conjecture, Math. Programming, to appear.Google Scholar
  6. 6.
    Chadder N., Deza A.: Computational determination of the largest lattice polytope diameter, in: Proceedings of the IX Latin and American Algorithms, Graphs, and Optimization Symposium. Electronic Notes in Discrete Mathematics 62, 105–110 (2017)CrossRefGoogle Scholar
  7. 7.
    Del Pia A., Michini C.: On the diameter of lattice polytopes. Discrete Comput. Geom 55, 681–687 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Deza, G. Manoussakis and S. Onn, Primitive zonotopes, Discrete Comput. Geom., to appear.Google Scholar
  9. 9.
    M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer (1993).Google Scholar
  10. 10.
    Kalai G., Kleitman D.: A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Amer. Math. Soc., 26, 315–316 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kleinschmidt P., Onn S.: On the diameter of convex polytopes. Discrete Math., 102, 75–77 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Naddef D.: The Hirsch conjecture is true for (0,1)-polytopes. Math. Programming 45, 109–110 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Naddef D., Pulleyblank W.: Hamiltonicity in (0,1)-polyhedra. J. Combin.Theory B, 37, 41–52 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Santos F.: A counterexample to the Hirsch conjecture. Ann. of Math., 176, 383–412 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sukegawa N.: Improving bounds on the diameter of a polyhedron in high dimensions. Discrete Math., 340, 2134–2142 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    T. Thiele, Extremalprobleme für Punktmengen, Diplomarbeit, Freie Universität Berlin (1991).Google Scholar
  17. 17.
    Todd M.: An improved Kalai–Kleitman bound for the diameter of a polyhedron. SIAM J. Discrete Math., 28, 1944–1947 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Springer (1995).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.McMaster UniversityOntarioCanada
  2. 2.Université de Paris SudOrsayFrance
  3. 3.LIPNVilletaneuseFrance

Personalised recommendations