Given a polyhedron we write P(I) for the convex hull of the integral points in P. It is know that P(I) can have at most O(fi(n-1)) vertices if P is a rational polyhedron with size fi. Here we give an example showing that P(I) can have as many as Omega (fi(n-1)) vertices. The construction uses...

Given a generic m x n matrix A, the simplicial complex {bold K}(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax = b}. The main result of this paper is that the topological space associated with {bold K}(A) is homeomorphic...

Given a generic m by n matrix A, a lattice point h in {bold Z} is a neighbor of the origin if the body {x : Ax = b}, with b_{i} = max{0, a_{i}h}, i = 1, ..., m, contains no lattice point other than 0 and h. The set of neighbors, N(A), is finite and Asymmetric. We show that if A' is another...

Samle [S] has recently shown how to estimate the average number of pivot steps in Lemke's algorithm for the linear complementarity problem (LCP) in terms of the "volumes" of certain cones. In this paper we discuss the notion of average volumes of cones, and give two applications to the LCP.

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.