The simplicial complex K (A) is defined to be the collection of simplices, and their proper
subsimplices, representing maximal lattice free bodies of the form (x: Ax⩽ b), with A a fixed
generic (n+ 1)× n matrix. The topological space associated with K (A) is shown to be
homeomorphic to ℝ n, and the space obtained by identifying lattice translates of these
simplices is homeorphic to the n-torus.