Improved bounds on the diameter of lattice polytopes |
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Authors: | A Deza L Pournin |
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Institution: | 1.McMaster University,Ontario,Canada;2.Université de Paris Sud,Orsay,France;3.LIPN,Villetaneuse,France |
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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 \({kd - \lceil2d/3\rceil-(k-3)}\) 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. |
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