3,141 research outputs found
On the complexity of the set of unconditional convex bodies
We show that for any , the set of unconditional convex bodies in
contains a -separated subset of cardinality at least . This implies that there exists an unconditional convex body in
which cannot be approximated within the distance by a
projection of a polytope with faces unless . We also show
that for , the cardinality of a -separated set of completely symmetric
bodies in does not exceed .Comment: 19 page
Reverse and dual Loomis-Whitney-type inequalities
Various results are proved giving lower bounds for the th intrinsic volume
, , of a compact convex set in , in
terms of the th intrinsic volumes of its projections on the coordinate
hyperplanes (or its intersections with the coordinate hyperplanes). The bounds
are sharp when and . These are reverse (or dual, respectively)
forms of the Loomis-Whitney inequality and versions of it that apply to
intrinsic volumes. For the intrinsic volume , which corresponds to mean
width, the inequality obtained confirms a conjecture of Betke and McMullen made
in 1983
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
The isotropic constant of random polytopes with vertices on convex surfaces
For an isotropic convex body we consider the isotropic
constant of the symmetric random polytope generated by
independent random points which are distributed according to the cone
probability measure on the boundary of . We show that with overwhelming
probability , where is an
absolute constant. If is unconditional we argue that even
with overwhelming probability. The proofs are based on concentration
inequalities for sums of sub-exponential or sub-Gaussian random variables,
respectively, and, in the unconditional case, on a new -estimate for
linear functionals with respect to the cone measure in the spirit of Bobkov and
Nazarov, which might be of independent interest
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