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Improved bounds on the diameter of lattice polytopes

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arxiv 1610.00341 v1 pith:OC4DUPNQ submitted 2016-10-02 math.MG math.COmath.OC

Antoine Deza , Lionel Pournin This is my paper · ORCID
classification math.MGmath.COmath.OC
keywords deltadiameterlatticeachievedadditionboundsconjecturecoordinates
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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 $kd-\lceil2d/3\rceil$ 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.

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