CONVEX LATTICE POLYTOPES

A lattice polytope is a polytope in E('n), all of whose vertices are lattice points. In this thesis, properties of convex lattice polytopes are investigated. Discrete analogs of Minkowski's theorem are found. Inequalities are derived relating the number of interior lattice points with various parameters associated with the polytope. Discrete isoperimetric inequalities are also studied.;Other results are proven with the aid of a computer. For example, a convex lattice decagon must have diameter at least SQRT.(29).;Integral unimodular affine transformations are used to charac- terize convex lattice polygons with precisely 1 interior lattice point.;A polygon is said to be realizable in the lattice if there is a lattice polygon similar to the given polygon. Realization theorems are found. For example, a convex polygon, K, is realizable if and only if for some triangulation of K by diagonals, the angles of each triangle so formed have rational tangents.;Examples. If a convex lattice polygon has width greater than SQRT.((2g+1)SQRT.(3)) then it must have at least g interior lattice points. If the polygon has 2n + 1 vertices, then it must have at least 3n - 5 interior lattice points.;Combinatorial properties are also studied. As an example, the following Ramsey-like results is obtained. Two lattice points x and y are said to form a hole in a set S if there is some lattice point between x and y that is not in S. If S is a set of m('n) + 1 lattice points in E('n), then either some 2 points of S form a hole, or some m + 1 points of S colline.