Matroid polytope

In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid.

, the matroid polytope

is the convex hull of the indicator vectors of the bases of

be a matroid on

, the indicator vector of

th unit vector in

The matroid polytope

is the convex hull of the set The matroid independence polytope or independence matroid polytope is the convex hull of the set The (basis) matroid polytope is a face of the independence matroid polytope.

ψ

, the independence matroid polytope is equal to the polymatroid determined by

The flag matroid polytope is another polytope constructed from the bases of matroids.

is a strictly increasing sequence of finite sets.

be the cardinality of the set

are said to be concordant if their rank functions satisfy Given pairwise concordant matroids

on the ground set

, consider the collection of flags

is a basis of the matroid

Such a collection of flags is a flag matroid

are called the constituents of

in a flag matroid

be the sum of the indicator vectors of each basis in

Given a flag matroid

, the flag matroid polytope

is the convex hull of the set A flag matroid polytope can be written as a Minkowski sum of the (basis) matroid polytopes of the constituent matroids:[4]