The complex of maximal lattice free simplices (Q1337134)

A maximal lattice free body is a body of the form \(K= \{x\mid Ax\leq b\}\) containing no lattice points in its interior such that any closed convex body which properly contains \(K\) does have a lattice point in its interior. The simplicial complex \(K(A)\) is defined to be the collection of simplices and their proper subsimplices representing maximal lattice free bodies of the above form with \(A\) a fixed generic \((n+ 1)\times n\)- matrix. The topological space associated with \(K(A)\) is shown to be homeomorphic to \(\mathbb{R}^ n\). The complex \(KZ(A)\) is defined by identifying those simplices or subsimplices of \(K(A)\) which are lattice translates of each other. The associated space is homeomorphic to the \(n\)-torus.