New inequalities for subspace arrangements

Abstract

For each positive integer $n\ge 4$, we give an inequality satisfied by rank functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's 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.

10 pages, comments welcome. v2: correction to proof of Prop. 3, improved "Future directions" section, other minor improvements. v3: final version, minor changes

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