Volúmenes de politopos matroidales

Abstract

In this work, the volume of matroid polytopes is studied through the properties of generalized permutohedra. Initially, a detailed contextualization of polytopes is presented through convex sets. Next, an algebraic structure for polytopes is provided via the Minkowski sum and the convex hull of two or more polytopes. Subsequently, the definition of a matroid is approached from the perspective of independent sets and its equivalence in terms of its bases, which is highly relevant as it allows the rank of a matroid to be defined. Some properties of matroids are mentioned, and the beta invariant of a matroid is described—a key tool for defining the volume of a matroid polytope, which arises when defining a polytope over a set of vectors associated with the base of a matroid. Likewise, generalized permutohedra are discussed, starting from the usual permutohedron. Additionally, the concept of a mixed volume over a convex body is described. Throughout the work, various examples and graphs are presented to visually illustrate the concepts and results discussed in the paper and found in the bibliography.