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On integer points in polyhedra: A lower bound

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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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Authors and Affiliations

  1. Mathematical Institute, Pf. 127, 1364, Budapest, Hungary

    Imre Bárány

  2. Department of Mathematics, Yale University, 06520, New Haven, CT, U.S.A.

    Roger Howe

  3. Department of Computer Science, Eötvös University, Múzeum krt. 6-8, 1088, Budapest, Hungary

    László Lovász

Authors
  1. Imre Bárány
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  2. Roger Howe
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  3. László Lovász
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Corresponding author

Correspondence to László Lovász.

Additional information

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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  • DOI: https://doi.org/10.1007/BF01204716

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