I have made a study of polytopes with the books of Ziegler and "Integer Programming" of Conforti, my main goal is to study **matroid polytopes**; to study matroids I have thought about the book "MATROID THEORY" of Welsh. But how do I connect the two topics, i.e. polytopes with matroids? I am not clear if in the same text of Welsh this is specified, or if there is any additional reference to do this work? Any suggestions are welcome!

## 2 Answers

You might like the book *Coxeter Matroids* by Borovik, Gelfand, and White.

In some sense, this book is actually about generalizations of matroids. One generalization is from matroids to *flag matroids* (which are sequences of matroids related by quotient/concordance). The other direction of generalization is to other Coxeter types. Usual matroids are type A (and related to the permutohedron stuff mentioned in the comment by Sam Hopkins).

While many books will be completely dedicated to matroid via various axioms, this book already introduces the polytope perspective in the first chapter.

Matroid polytopes are a standard topic in the field of combinatorial optimization, and as such I would recommend the beautiful text Combinatorial Optimization - Polyhedra and Efficiency by Lex Schrijver. This is an encyclopedic treatment of the subject covering far more than matroid polytopes. Matroid polytopes (and many other topics related to matroids) are covered in Part IV. For example, Edmonds' classic inequality description of the independent set polytope of a matroid is presented in Part IV.

6more comments