Matroid base polytope decomposition II: Sequences of hyperplane splits

This is a continuation of an early paper Chatelain et al. (2011) [3] about matroid base polytope decomposition. We will present sufficient conditions on a matroid M so its base polytope P(M) has a sequence of hyperplane splits. These yield to decompositions of P(M) with two or more pieces for infinitely many matroids M. We also present necessary conditions on the Euclidean representation of rank three matroids M for the existence of decompositions of P(M) into 2 or 3 pieces. Finally, we prove that P(M1⊕M2) has a sequence of hyperplane splits if either P(M1) or P(M2) also has a sequence of hyperplane splits.