A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is eq...

2 Convex Polyhedron 3 2.1 What is convex polytope/polyhedron? . . . . . . . . . . . . . . . . . . . . . . 3 2.2 What are the faces of a convex polytope/polyhedron? . . . . . . . . . . . . . . 3 2.3 What is the face lattice of a convex polytope . . . . . . . . . . . . . . . . . . . 4 2.4 What is a dual of a convex polytope? . . . . . . . . . . . . . . . . . . . . . . . 4 2.5 What is simplex? . ....

We investigate geometric and topological properties of $d$-majorization -- a generalization classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive new, simplified characterization which allows us work out halfspace description the corresponding polytopes. That is, write set all are $d$-majorized by some given vector $y \mathbb{R}^n$ as an intersection f...

We consider three well-studied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an...