On integer points in polyhedra: A lower bound

Abstract

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

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Correspondence to László Lovász.

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

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Bárány, I., Howe, R. & Lovász, L. On integer points in polyhedra: A lower bound. Combinatorica 12, 135–142 (1992). https://doi.org/10.1007/BF01204716

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AMS subject classification code (1991)

  • 52 C 07
  • 11 H 06