2,572 research outputs found
New inequalities for subspace arrangements
For each positive integer , we give an inequality satisfied by rank
functions of arrangements of subspaces. When we recover Ingleton's
inequality; for higher 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.Comment: 10 pages, comments welcome. v2: correction to proof of Prop. 3,
improved "Future directions" section, other minor improvements. v3: final
version, minor change
Obstructions to determinantal representability
There has recently been ample interest in the question of which sets can be
represented by linear matrix inequalities (LMIs). A necessary condition is that
the set is rigidly convex, and it has been conjectured that rigid convexity is
also sufficient. To this end Helton and Vinnikov conjectured that any real zero
polynomial admits a determinantal representation with symmetric matrices. We
disprove this conjecture. By relating the question of finding LMI
representations to the problem of determining whether a polymatroid is
representable over the complex numbers, we find a real zero polynomial such
that no power of it admits a determinantal representation. The proof uses
recent results of Wagner and Wei on matroids with the half-plane property, and
the polymatroids associated to hyperbolic polynomials introduced by Gurvits.Comment: 10 pages. To appear in Advances in Mathematic
On densities of lattice arrangements intersecting every i-dimensional affine subspace
In 1978, Makai Jr. established a remarkable connection between the
volume-product of a convex body, its maximal lattice packing density and the
minimal density of a lattice arrangement of its polar body intersecting every
affine hyperplane. Consequently, he formulated a conjecture that can be seen as
a dual analog of Minkowski's fundamental theorem, and which is strongly linked
to the well-known Mahler-conjecture.
Based on the covering minima of Kannan & Lov\'asz and a problem posed by
Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and
investigate densities of lattice arrangements of convex bodies intersecting
every i-dimensional affine subspace. Then it becomes natural also to formulate
and study a dual analog to Minkowski's second fundamental theorem. As our main
results, we derive meaningful asymptotic lower bounds for the densities of such
arrangements, and furthermore, we solve the problems exactly for the special,
yet important, class of unconditional convex bodies.Comment: 19 page
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
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