9,775 research outputs found
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
On Metric Ramsey-type Dichotomies
The classical Ramsey theorem, states that every graph contains either a large
clique or a large independent set. Here we investigate similar dichotomic
phenomena in the context of finite metric spaces. Namely, we prove statements
of the form "Every finite metric space contains a large subspace that is nearly
quilateral or far from being equilateral". We consider two distinct
interpretations for being "far from equilateral". Proximity among metric spaces
is quantified through the metric distortion D. We provide tight asymptotic
answers for these problems. In particular, we show that a phase transition
occurs at D=2.Comment: 14 pages, 0 figure
On some low distortion metric Ramsey problems
In this note, we consider the metric Ramsey problem for the normed spaces
l_p. Namely, given some 1=1, and an integer n, we ask
for the largest m such that every n-point metric space contains an m-point
subspace which embeds into l_p with distortion at most alpha. In
[arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of
on alpha undergoes a phase transition at alpha=2. Here we consider this
problem for other l_p, and specifically the occurrence of a phase transition
for p other than 2. It is shown that a phase transition does occur at alpha=2
for every p in the interval [1,2]. For p>2 we are unable to determine the
answer, but estimates are provided for the possible location of such a phase
transition. We also study the analogous problem for isometric embedding and
show that for every 1<p<infinity there are arbitrarily large metric spaces, no
four points of which embed isometrically in l_p.Comment: 14 pages, to be published in Discrete and Computational Geometr
Limitations to Frechet's Metric Embedding Method
Frechet's classical isometric embedding argument has evolved to become a
major tool in the study of metric spaces. An important example of a Frechet
embedding is Bourgain's embedding. The authors have recently shown that for
every e>0 any n-point metric space contains a subset of size at least n^(1-e)
which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is
non-Frechet, and the purpose of this note is to show that this is not
coincidental. Specifically, for every e>0, we construct arbitrarily large
n-point metric spaces, such that the distortion of any Frechet embedding into
l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur
Metric Cotype
We introduce the notion of metric cotype, a property of metric
spaces related to a property of normed spaces, called Rademacher
cotype. Apart from settling a long standing open problem in metric
geometry, this property is used to prove the following dichotomy: A
family of metric spaces F is either almost universal (i.e., contains
any finite metric space with any distortion > 1), or there exists
α > 0, and arbitrarily large n-point metrics whose distortion when
embedded in any member of F is at least Ω((log n)^α). The same
property is also used to prove strong non-embeddability theorems
of L_q into L_p, when q > max{2,p}. Finally we use metric cotype
to obtain a new type of isoperimetric inequality on the discrete
torus
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