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 (n + 1) × n matrix. The topological space associated with K(A) is shown to be homeomorphic to Rn, and the space obtained by identifying lattice translates of these simplices is homeomorphic to the n-torus.
Given a polyhedron P subset Rn we write PI for the convex hull of the integral points in P. It is known that PI can have at most O(ϕn-1) vertices if P is a rational polyhedron with size ϕ. Here we give an example showing that PI can have as many as Ω(ϕn-1) vertices. The construction uses the Dirichlet unit theorem.
Keywords: Polyhedra; integral points, Dirichlet unit theorem
It is suggested that the ethical notion of social contract can be formally modeled using the well-studied concept of the core of a game. This provides a mathematical technique for studying social contracts and theories of justice. The idea is applied to Rawlsian justice here.