Dmitry V. Gribanov

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Let A be an (m × n) integral matrix, and let P = {x : Ax ≤ b} be an n-dimensional polytope. The width of P is defined as w(P) = min{x ∈ Z n \ {0} : max x∈P x ⊤ u − min x∈P x ⊤ v}. Let ∆(A) and δ(A) denote the greatest and the smallest absolute values of a determinant among all r(A) × r(A) sub-matrices of A, where r(A) is the rank of a matrix A. We prove(More)
Аннотация Let A be an (m × n) integral matrix of the rank n and let P = {x : Ax ≤ b} be n-dimensional polytope. Width of P is defined as w(P) = min{maxP x T u − minP x T v : x ∈ Z n \ {0}}. Let ∆ denote the smallest absolute value of the determinant among basis matrices of A. We prove that if every basis matrix of A has determinant equal to ±∆ and w(P) ≥ (∆(More)
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