Polytopes

Abstract

Our basic datum is a list X := (a₁, …, a_m) of nonzero elements in a real s-dimensional vector space V (we allow repetitions in the list, since this is important for the applications). Sometimes we take $V = \mathbb{R}^{s}$ and then think of the a_i as the columns of an $s \times m$matrix A.

From X we shall construct several geometric, algebraic, and combinatorial objects such as the cone they generate, a hyperplane arrangement, some families of convex polytopes, certain algebras and modules, and some special functions. Our aim is to show how these constructions relate to each other, and taken together, they provide structural understanding and computational techniques.