18,960 research outputs found
Measured descent: A new embedding method for finite metrics
We devise a new embedding technique, which we call measured descent, based on
decomposing a metric space locally, at varying speeds, according to the density
of some probability measure. This provides a refined and unified framework for
the two primary methods of constructing Frechet embeddings for finite metrics,
due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space
(X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where
alpha_X is a geometric estimate on the decomposability of X. As an immediate
corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding,
where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this
result recovers Bourgain's theorem, but when the metric X is, in a sense,
``low-dimensional,'' improved bounds are achieved.
Our embeddings are volume-respecting for subsets of arbitrary size. One
consequence is the existence of (k, O(log n)) volume-respecting embeddings for
all 1 \leq k \leq n, which is the best possible, and answers positively a
question posed by U. Feige. Our techniques are also used to answer positively a
question of Y. Rabinovich, showing that any weighted n-point planar graph
embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the
dimension is optimal, and improves upon the previously known bound of O((log
n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric &
Functional Analysis. This version fixes a subtle error in Section 2.
Sublinear Higson corona and Lipschitz extensions
The purpose of the paper is to characterize the dimension of sublinear Higson
corona of in terms of Lipschitz extensions of functions:
Theorem: Suppose is a proper metric space. The dimension of the
sublinear Higson corona of is the smallest integer with
the following property: Any norm-preserving asymptotically Lipschitz function
, , extends to a norm-preserving
asymptotically Lipschitz function .
One should compare it to the result of Dranishnikov \cite{Dr1} who
characterized the dimension of the Higson corona of is the
smallest integer such that is an absolute extensor of
in the asymptotic category \AAA (that means any proper asymptotically
Lipschitz function , closed in , extends to a
proper asymptotically Lipschitz function ). \par
In \cite{Dr1} Dranishnikov introduced the category \tilde \AAA whose
objects are pointed proper metric spaces and morphisms are asymptotically
Lipschitz functions such that there are constants
satisfying
for all .
We show if and only if is an absolute
extensor of in the category \tilde\AAA. \par As an application we reprove
the following result of Dranishnikov and Smith \cite{DRS}:
Theorem: Suppose is a proper metric space of finite asymptotic
Assouad-Nagata dimension \asdim_{AN}(X). If is cocompact and connected,
then \asdim_{AN}(X) equals the dimension of the sublinear Higson corona
of .Comment: 13 page
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