130 research outputs found
Compression bounds for Lipschitz maps from the Heisenberg group to
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg
group with its Carnot-Carath\'eodory metric and apply it to give a lower bound
on the integrality gap of the Goemans-Linial semidefinite relaxation of the
Sparsest Cut problem
Vertical versus horizontal Poincar\'e inequalities on the Heisenberg group
Let be the discrete
Heisenberg group, equipped with the left-invariant word metric
associated to the generating set .
Letting B_n= {x\in \H: d_W(x,e_\H)\le n} denote the corresponding closed ball
of radius , and writing , we prove that if
is a Banach space whose modulus of uniform convexity has power
type then there exists such that every
satisfies {multline*} \sum_{k=1}^{n^2}\sum_{x\in
B_n}\frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}}\le K\sum_{x\in B_{21n}}
\Big(|f(xa)-f(x)|^q_X+\|f(xb)-f(x)\|^q_X\Big). {multline*} It follows that for
every the bi-Lipschitz distortion of every is at least a
constant multiple of , an asymptotically optimal estimate as
A doubling subset of for that is inherently infinite dimensional
It is shown that for every there exists a doubling subset
of that does not admit a bi-Lipschitz embedding into for any
Bourgain's discretization theorem
Bourgain's discretization theorem asserts that there exists a universal
constant with the following property. Let be Banach
spaces with . Fix and set .
Assume that is a -net in the unit ball of and that
admits a bi-Lipschitz embedding into with distortion at most
. Then the entire space admits a bi-Lipschitz embedding into with
distortion at most . This mostly expository article is devoted to a
detailed presentation of a proof of Bourgain's theorem.
We also obtain an improvement of Bourgain's theorem in the important case
when for some : in this case it suffices to take
for the same conclusion to hold true. The case
of this improved discretization result has the following consequence. For
arbitrarily large there exists a family of
-point subsets of such that if we write
then any embedding of , equipped with the
Earthmover metric (a.k.a. transportation cost metric or minimumum weight
matching metric) incurs distortion at least a constant multiple of
; the previously best known lower bound for this problem was
a constant multiple of .Comment: Proof of Lemma 5.1 corrected; its statement remains unchange
Markov convexity and nonembeddability of the Heisenberg group
We compute the Markov convexity invariant of the continuous infinite
dimensional Heisenberg group to show that it is Markov
4-convex and cannot be Markov -convex for any . As Markov convexity
is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a
different proof of the classical theorem of Pansu and Semmes that the
Heisenberg group does not admit a biLipschitz embedding into any Euclidean
space.
The Markov convexity lower bound will follow from exhibiting an explicit
embedding of Laakso graphs into that has distortion
at most . We use this to show that if is a Markov
-convex metric space, then balls of the discrete Heisenberg group
of radius embed into with distortion at least
some constant multiple of
Finally, we show that Markov 4-convexity does not give the optimal distortion
for embeddings of binary trees into by showing that
the distortion is on the order of .Comment: version to appear in Ann. Inst. Fourie
- …