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 b1 = max {0, aih }, 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 i h = 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 i h