, Volume 12, Issue 2, pp 135-142

On integer points in polyhedra: A lower bound

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Abstract

Given a polyhedronP⊂ℝ we writeP I for the convex hull of the integral points inP. It is known thatP I can have at most135-2 vertices ifP is a rational polyhedron with size φ. Here we give an example showing thatP I can have as many as Ω(ϕ n−1) vertices. The construction uses the Dirichlet unit theorem.

The results of the paper were obtained while this author was visiting the Cowles Foundation at Yale University