Matroid base polytope decomposition

Let P(M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P(M) is a decomposition of the form where each P(Mi) is also a matroid base polytope for some matroid Mi, and for each 1⩽i≠j⩽t, the intersection P(Mi)∩P(Mj) is a face of both P(Mi) and P(Mj). In this paper, we investigate hyperplane splits, that is, polytope decompositions when t=2. We give sufficient conditions for M so P(M) has a hyperplane split and characterize when P(M1⊕M2) has a hyperplane split where M1⊕M2 denote the direct sum of matroids M1 and M2. We also prove that P(M) has not a hyperplane split if M is binary. Finally, we show that P(M) has not a decomposition if its 1-skeleton is the hypercube.