On some combinatorial properties of matroid base polytope

We will begin by recalling a well-understood hierarchy of lattice polytope properties. Some of these properties translate into algebraic properties of the corresponding toric ideals. For instance, a polytope possesses a quadratic regular unimodular triangulation if and only if its corresponding toric ideal has a quadratic Groebner basis. The existence of such a triangulation is an open problem for matroid base polytopes, but recently a weaker property was shown resolving a conjecture of Cunningham from 1984.