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The Structure of the Complex of Maximal Lattice Free Bodies for a Matrix of Size (n + 1) × n

In: Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research

Author

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  • Herbert E. Scarf

    (Yale University)

Abstract

The complex of maximal lattice free bodies associated with a well behaved matrix A of size (n + 1) × n is generated by a finite set of simplicies, K0(A), of the form {0, h1, …, hk},with k ≤ n, and their lattice translates. The simplicies in K0(A) are selected so that the plane a0x = 0, with ao the first row of A, passes through the vertex 0. The collection of simplicies {h1,…,hk} is denoted by Top. Various properties of Top are demonstrated, including the fact that no two interior faces of Top are lattice translates of each other. Moreover, if g is a generator of the cone generated by the set of neighbors {h} with a0h> 0, then the set of simplicies of Top which contain g is the union of linear intervals of simplicies with special features. These features lead to an algorithm for calculating the simplicies in K0(A)as a0 varies and the plane a0x = 0 passes through the generator ɡ.

Suggested Citation

  • Herbert E. Scarf, 2008. "The Structure of the Complex of Maximal Lattice Free Bodies for a Matrix of Size (n + 1) × n," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 13, pages 227-258, Palgrave Macmillan.
    • Handle: RePEc:pal:palchp:978-1-137-02441-1_13
    DOI: 10.1057/9781137024411_13
    as

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