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Mathematics  2009 

New inequalities for subspace arrangements

DOI: 10.1016/j.jcta.2009.10.014

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Abstract:

For each positive integer $n \geq 4$, 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.

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