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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
Article

Abstract

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 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

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

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