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Distance between vertices of lattice polytopes

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Abstract

A lattice (d,k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers ranging between 0 and k. We consider the largest possible distance \(\delta \)(d,k) between two vertices in the edge-graph of a lattice (d,k)-polytope. We show that \(\delta \)(5,3) and \(\delta \)(3,6) are equal to 10. This substantiates the conjecture whereby \(\delta \)(d,k) is achieved by a Minkowski sum of lattice vectors.

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Change history

  • 30 January 2020

    The erratum mostly concerns Table 4 and Figure 6 where two polytopes were misrepresented in the original version.

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Acknowledgements

The authors thank the anonymous referees for their valuable comments. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant program (RGPIN-2015-06163), by the Exceptional Opportunity program (ESROP), and by the ANR project SoS (Structures on Surfaces) ANR-17-CE40-0033.

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Correspondence to Antoine Deza.

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Deza, A., Deza, A., Guan, Z. et al. Distance between vertices of lattice polytopes. Optim Lett 14, 309–326 (2020). https://doi.org/10.1007/s11590-018-1338-7

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Keywords

  • Lattice polytope
  • Diameter
  • Minkowski sum