348 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
Low-distortion embeddings of graphs with large girth
The main purpose of the paper is to construct a sequence of graphs of
constant degree with indefinitely growing girths admitting embeddings into
with uniformly bounded distortions. This result answers the problem
posed by N. Linial, A. Magen, and A. Naor (2002).Comment: Some confusing omissions are corrected in the second versio
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
Fixed point property for a CAT(0) space which admits a proper cocompact group action
We prove that if a geodesically complete space admits a
proper cocompact isometric action of a group, then the Izeki-Nayatani invariant
of is less than . Let be a finite connected graph, be
the linear spectral gap of , and be the nonlinear spectral
gap of with respect to such a space . Then, the result
implies that the ratio is bounded from below by a
positive constant which is independent of the graph . It follows that any
isometric action of a random group of the graph model on such has a global
fixed point. In particular, any isometric action of a random group of the graph
model on a Bruhat-Tits building associated to a semi-simple algebraic group has
a global fixed point
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