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arXiv · 1610.00341

Improved bounds on the diameter of lattice polytopes

Antoine Deza·Lionel Pournin
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Abstract

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

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Antoine Deza, Lionel Pournin. 2016-10-02. Improved bounds on the diameter of lattice polytopes. https://doi.org/10.1007/s10474-017-0777-4

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