User:Dchaws/Matroid Polytope

Matroid polytopes are closely related to polymatroids. Given a base $$B \subseteq \{1,\ldots,n\}$$ of a matroid $$M$$ on $$n$$ elements, we define the incidence vector of $$ B $$ as $$ \mathbf e_B := \sum_{i \in B}^n \mathbf e_i $$ where $$ \mathbf e_i $$ are the standard unit vectors in $$ \mathbf{R}^n $$. The matroid polytope of $$M$$ is the convex hull of the set $$\{\, \mathbf e_B \mid B \text{ a base of } M \,\} \subseteq \mathbf R^n$$. This is also referred to as the matroid base polytope.

Independence Matroid Polytope
The independence matroid polytope is the convex hull of the set $$\{\, \mathbf e_I \mid I \text{ an independent set of } M \,\} \subseteq \mathbf R^n$$. The matroid polytope is a face of the independence matroid polytope. Given the rank $$ \psi $$ of a matroid $$ M $$, the independence matroid polytope is equal to the polymatroid determined by $$ \psi $$.