Abstract.
The d-step conjecture is one of the fundamental open problems concerning the structure of convex polytopes. Let Δ (d,n) denote the maximum diameter of a graph of a d-polytope that has n facets. The d-step conjecture Δ (d,2d) = d is proved equivalent to the following statement: For each ``general position'' \((d-1)\times (d-1)\) real matrix M there are two matrices \(Q_{\tau}, Q_{\sigma}\) drawn from a finite group \(\hat{S}_d$ of $(d-1)\times (d-1)\) matrices isomorphic to the symmetric group \(\mathop{\rm Sym}\nolimits (d)\) on d letters, such that \(Q_{\tau} MQ_{\sigma}\) has the Gaussian elimination factorization L -1 U in which L and U are lower triangular and upper triangular matrices, respectively, that have positive nontriangular elements. If #(M) is the number of pairs \((\sigma,\tau) \in \mathop{\rm Sym}\nolimits(d) \times \mathop{\rm Sym}\nolimits (d)\) giving a positive L -1 U factorization, then #(M) equals the number of d-step paths between two vertices of an associated Dantzig figure. One consequence is that #(M)≤ d!. Numerical experiments all satisfied #(M) ≥ 2 d-1, including examples attaining equality for 3 ≤ d ≤ 15. The inequality #(M) ≥ 2 d-1 is proved for d=3. For d≥ 4, examples with #(M) =2 d-1 exhibit a large variety of combinatorial types of associated Dantzig figures. These experiments and other evidence suggest that the d-step conjecture may be true in all dimensions, in the strong form #(M) ≥ 2 d-1.
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Received April 10, 1995, and in revised form August 23, 1995.
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Lagarias, J., Prabhu, N. & Reeds, J. The d-Step Conjecture and Gaussian Elimination . Discrete Comput Geom 18, 53–82 (1997). https://doi.org/10.1007/PL00009308
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DOI: https://doi.org/10.1007/PL00009308