101,468 research outputs found
A new notion of angle between three points in a metric space
We give a new notion of angle in general metric spaces; more precisely, given
a triple a points in a metric space , we introduce the notion of
angle cone as being an interval
, where the quantities
are defined in terms of the distance functions from and
via a duality construction of differentials and gradients holding for
locally Lipschitz functions on a general metric space. Our definition in the
Euclidean plane gives the standard angle between three points and in a
Riemannian manifold coincides with the usual angle between the geodesics, if
is not in the cut locus of or . We show that in general the angle
cone is not single valued (even in case the metric space is a smooth Riemannian
manifold, if is in the cut locus of or ), but if we endow the metric
space with a positive Borel measure obtaining the metric measure space
then under quite general assumptions (which include many fundamental
examples as Riemannian manifolds, finite dimensional Alexandrov spaces with
curvature bounded from below, Gromov-Hausdorff limits of Riemannian manifolds
with Ricci curvature bounded from below, and normed spaces with strictly convex
norm), fixed , the angle cone at is single valued for -a.e.
. We prove some basic properties of the angle cone (such as the
invariance under homotheties of the space) and we analyze in detail the case
is a measured-Gromov-Hausdorff limit of a sequence of Riemannian
manifolds with Ricci curvature bounded from below, showing the consistency of
our definition with a recent construction of Honda.Comment: 19 page
Spectral geometry with a cut-off: topological and metric aspects
Inspired by regularization in quantum field theory, we study topological and
metric properties of spaces in which a cut-off is introduced. We work in the
framework of noncommutative geometry, and focus on Connes distance associated
to a spectral triple (A, H, D). A high momentum (short distance) cut-off is
implemented by the action of a projection P on the Dirac operator D and/or on
the algebra A. This action induces two new distances. We individuate conditions
making them equivalent to the original distance. We also study the
Gromov-Hausdorff limit of the set of truncated states, first for compact
quantum metric spaces in the sense of Rieffel, then for arbitrary spectral
triples. To this aim, we introduce a notion of "state with finite moment of
order 1" for noncommutative algebras. We then focus on the commutative case,
and show that the cut-off induces a minimal length between points, which is
infinite if P has finite rank. When P is a spectral projection of , we work
out an approximation of points by non-pure states that are at finite distance
from each other. On the circle, such approximations are given by Fejer
probability distributions. Finally we apply the results to Moyal plane and the
fuzzy sphere, obtained as Berezin quantization of the plane and the sphere
respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2
figures. Journal of Geometry and Physics 201
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
We study the geometry of nonrelatively hyperbolic groups. Generalizing a
result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic
space in a relatively hyperbolic space is contained in a bounded neighborhood
of a single peripheral subgroup. This implies that a group being relatively
hyperbolic with nonrelatively hyperbolic peripheral subgroups is a
quasi-isometry invariant. As an application, Artin groups are relatively
hyperbolic if and only if freely decomposable.
We also introduce a new quasi-isometry invariant of metric spaces called
metrically thick, which is sufficient for a metric space to be nonhyperbolic
relative to any nontrivial collection of subsets. Thick finitely generated
groups include: mapping class groups of most surfaces; outer automorphism
groups of most free groups; certain Artin groups; and others. Nonuniform
lattices in higher rank semisimple Lie groups are thick and hence nonrelatively
hyperbolic, in contrast with rank one which provided the motivating examples of
relatively hyperbolic groups. Mapping class groups are the first examples of
nonrelatively hyperbolic groups having cut points in any asymptotic cone,
resolving several questions of Drutu and Sapir about the structure of
relatively hyperbolic groups. Outside of group theory, Teichmuller spaces for
surfaces of sufficiently large complexity are thick with respect to the
Weil-Peterson metric, in contrast with Brock--Farb's hyperbolicity result in
low complexity.Comment: To appear in Mathematische Annale
Center-based Clustering under Perturbation Stability
Clustering under most popular objective functions is NP-hard, even to
approximate well, and so unlikely to be efficiently solvable in the worst case.
Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at
bypassing this computational barrier by using properties of instances one might
hope to hold in practice. In particular, they argue that instances in practice
should be stable to small perturbations in the metric space and give an
efficient algorithm for clustering instances of the Max-Cut problem that are
stable to perturbations of size . In addition, they conjecture that
instances stable to as little as O(1) perturbations should be solvable in
polynomial time. In this paper we prove that this conjecture is true for any
center-based clustering objective (such as -median, -means, and
-center). Specifically, we show we can efficiently find the optimal
clustering assuming only stability to factor-3 perturbations of the underlying
metric in spaces without Steiner points, and stability to factor
perturbations for general metrics. In particular, we show for such instances
that the popular Single-Linkage algorithm combined with dynamic programming
will find the optimal clustering. We also present NP-hardness results under a
weaker but related condition
Finite Volume Spaces and Sparsification
We introduce and study finite -volumes - the high dimensional
generalization of finite metric spaces. Having developed a suitable
combinatorial machinery, we define -volumes and show that they contain
Euclidean volumes and hypertree volumes. We show that they can approximate any
-volume with multiplicative distortion. On the other hand, contrary
to Bourgain's theorem for , there exists a -volume that on vertices
that cannot be approximated by any -volume with distortion smaller than
.
We further address the problem of -dimension reduction in the context
of volumes, and show that this phenomenon does occur, although not to
the same striking degree as it does for Euclidean metrics and volumes. In
particular, we show that any metric on points can be -approximated by a sum of cut metrics, improving
over the best previously known bound of due to Schechtman.
In order to deal with dimension reduction, we extend the techniques and ideas
introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of
graph Sparsification, and develop general methods with a wide range of
applications.Comment: previous revision was the wrong file: the new revision: changed
(extended considerably) the treatment of finite volumes (see revised
abstract). Inserted new applications for the sparsification technique
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