New inequalities for subspace arrangements

For each positive integer n ⩾ 4 , we give an inequality satisfied by rank functions of arrangements of n subspaces. When n = 4 we recover Ingletonʹs inequality; for higher n the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the “cone of realizable polymatroids” are also presented.