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The Shapes of Polyhedra

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Abstract

Let A be a real matrix of size (n+d+1)xn. We assume that all n x n submatrices of A are non-singular and define the condition number C = C(A) to be the ratio of the largest n x n subdeterminant of A to the smallest in absolute value. In addition we assume that there is a positive vector pi such that (pi)A = 0. This implies that for any b, the body K(b) = 'X such that AX 0, there exists a subset of the bodies K(b), of cardinality not larger than f(A) 1/2(log to the base 2 of (nC)/epsilon^{d}, such that every body is within epsilon of some member of the subset.

Suggested Citation

  • Herbert E. Scarf & R. Kannan & Laszlo Lovasz, 1988. "The Shapes of Polyhedra," Cowles Foundation Discussion Papers 883, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:883
    Note: CFP 753.
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d08/d0883.pdf
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    Cited by:

    1. I. Bárány & H. E. Scarf & D. Shallcross, 2008. "The topological structure of maximal lattice free convex bodies: The general case," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 11, pages 191-205, Palgrave Macmillan.
    2. Herbert E. Scarf & Shallcross, David F., 1991. "Shortest Integer Vectors," Cowles Foundation Discussion Papers 965, Cowles Foundation for Research in Economics, Yale University.
    3. William Cook & Thomas Rutherford & Herbert E. Scarf & David F. Shallcross, 1991. "An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming," Cowles Foundation Discussion Papers 990, Cowles Foundation for Research in Economics, Yale University.
    4. Robert Weismantel, 1998. "Test sets of integer programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(1), pages 1-37, February.

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