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...

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...

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) x n matrix. The topological space associated with K(A) is shown to be homeomorphic to R^{n}, and the...

The set of possible outcomes of a strongly ordinal bimatrix game is studied by imbedding each pair of possible payoffs as a point on the standard two-dimensional integral lattice. In particular, we count the number of different Pareto optimal sets of each cardinality; we establish asymptotic...

The main result of this paper is that given n red, n white, and n green points in the plane, it is possible to form n vertex-disjoint triangles Delta_{1},...,Delta_{n} in such a way that the Delta_{i} has one red, one white, and one green vertex for every i = 1,...,n and the intersection of...

Let P_{r} denote the convex hull of the integer points in the disc of radius r. We prove that the number of vertices of P_{r} is essentially r^{2/3} as r approaches infinity.

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...