Matrices with Identical Sets of Neighbors

Given a generic m by n matrix A, a lattice point h in ℤⁿ is a neighbor of the origin if the body {x: Ax ≤ b}, with b_i = max{0, a_ih}, i = 1, …, m, contains no lattice point other than 0 and h. The set of neighbors, N(A), is finite and 0-symmetric. We show that if A′ is another matrix of the same size with the property that sign a_ih = sign a_i′h for every i and every h ∈ N(A), then A′ has precisely the same set of neighbors as A. The collection of such matrices is a polyhedral cone, described by a finite set of linear inequalities, each such inequality corresponding to a generator of one of the cones C_i = pos{h ∈ N(A): a_ih < 0}. Computational experience shows that C_i has “few” generators. We demonstrate this in the first nontrivial case n = 3, m = 4.