Examples Of Metric Spaces Research Articles

Lipschitz retractions and complementation properties of Banach spaces

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F(X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points {0,1} such that no separable subset of M containing these points is a Lipschitz retract of M. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

Thick families of geodesics and differentiation

The differentiation theory of Lipschitz functions on metric spaces taking values in a Banach space with the Radon-Nikodým property (RNP), originally developed by Cheeger—Kleiner, has proven to be a powerful tool to prove non-bi-Lipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not bi-Lipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. Our first result is that any metric space containing a thick family of geodesics also contains a subset and a probability measure on that subset which satisfies a weakened form of RNP Lipschitz differentiability. A corollary is a new nonembeddability result. Our second main result is that, if the metric space is a non-RNP Banach space, a subset consisting of a thick family of geodesics can be constructed to satisfy true RNP differentiability. An intriguing question is whether this differentiation criterion, or some weakened form of it such as the one we prove in the first result, actually characterizes general metric spaces non-bi-Lipschitz embeddable into RNP Banach spaces.

Elements of non-Euclidean geometry in the formation of the concept of rectilinear placement of points in schoolchildren

The paper deals with issues of the metric geometry basics. In particular, the concept of rectilinear placement of points is considered, based on the axioms of the distance between two points of metric space. This approach allows forming a modern view of the property of straightness in the pupils. This paper analyzes the content of existing mathematics textbooks for general educational institutions to acquaintance of pupils with the elements of metric geometry. The first part of the paper provides information about the rectilinear placement of points; it can be used in Geometry lessons in the 7th – 9th grades. Set of linear functions are considered as examples of points of metric space. The similar work was done in the second part of the work for geometric material of the 10th – 11th grades. In addition, some simple examples of metric spaces that may be accessible to pupils of the relevant classes are discussed. The purpose of the work is gradually introduction of pupils to the elements of non-Euclidean geometries, to form a generalized notion of the distance between the points and rectilinear of their placement. The work can be used for Mathematics teaching at school and for retraining of teachers of Mathematics.

A generalized tetrahedral property

We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and prove that this generalized definition retains all the results of the original Tetrahedral Property proven by Portegies-Sormani: it provides a lower bound on the sliced filling volume and a lower bound on the volumes of balls. Thus sequences with uniform bounds on this Generalized Tetrahedral Property also have subsequences which converge in both the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat sense to the same non-collapsed and countably rectifiable limit space.

Metrics on Doubles as an Inverse Semigroup

For a metric space X we study metrics on the two copies of X. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup M(X). Our main result is that M(X) is an inverse semigroup. Therefore, one can define the \(C^*\)-algebra of this inverse semigroup, which is not necessarily commutative. If the Gromov–Hausdorff distance between two metric spaces, X and Y, is finite then their inverse semigroups M(X) and M(Y) (and hence their \(C^*\)-algebras) are isomorphic. We characterize the metrics that are idempotents, and give examples of metric spaces for which the semigroup M(X) (and the corresponding \(C^*\)-algebra) is commutative. We also describe the class of metrics determined by subsets of X in terms of the closures of the subsets in the Higson corona of X and the class of invertible metrics.

Measure rigidity of Ricci curvature lower bounds

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space.We start our investigation from the Euclidean case by proving that if a positive Radon measure m over Rd is such that (Rd,|⋅|,m) verifies a weaker variant of MCP, then its support spt(m) must be convex and m has to be absolutely continuous with respect to the relevant Hausdorff measure of spt(m). This result is then used as a starting point to investigate the rigidity of MCP in the metric framework.We introduce the new notion of reference measure for a metric space and prove that if (X,d,m) is essentially non-branching and verifies MCP, and μ is an essentially non-branching MCP reference measure for (spt(m),d), then m is absolutely continuous with respect to μ, on the set of points where an inversion plan exists. As a consequence, an essentially non-branching MCP reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured Gromov–Hausdorff convergence, provided an additional uniform bound holds.In the final part we present concrete examples of metric spaces with reference measures, both in smooth and non-smooth setting. The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group.

Embeddable box spaces of free groups

We generalize the construction of Arzhantseva, Guentner and Spakula of a box space of the free group which admits a coarse embedding into Hilbert space. We show that for a finitely generated free group, the box space corresponding to the derived \(m\)-series (for any integer \(m\ge 2\)) coarsely embeds into Hilbert space. This gives new examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have Yu’s property A.

Coarse differentiation and quantitative nonembeddability for Carnot groups

We give lower bound estimates for the macroscopic scale of coarse differentiability of Lipschitz maps from a Carnot group with the Carnot–Carathéodory metric (G,dcc) to a few different classes of metric spaces. Using this result, we derive lower bound estimates for quantitative nonembeddability of Lipschitz embeddings of G into a metric space (X,dX) if X is either an Alexandrov space with nonpositive or nonnegative curvature, a superreflexive Banach space, or another Carnot group that does not admit a biLipschitz homomorphic embedding of G. For the same targets, we can further give lower bound estimates for the biLipschitz distortion of every embedding f:B(n)→X, where B(n) is the ball of radius n of a finitely generated nonabelian torsion-free nilpotent group G. We also prove an analogue of Bourgain's discretization theorem for Carnot groups and show that Carnot groups have nontrivial Markov convexity. These give the first examples of metric spaces that have nontrivial Markov convexity but cannot biLipschitzly embed into Banach spaces of nontrivial Markov convexity.

Box spaces, group extensions and coarse embeddings into Hilbert space

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincare inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.

Assouad–Nagata dimension of locally finite groups and asymptotic cones

In this paper we study two problems concerning Assouad–Nagata dimension: (1) Is there a metric space of positive asymptotic Assouad–Nagata dimension such that all of its asymptotic cones are of Assouad–Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]). (2) Suppose G is a locally finite group with a proper left invariant metric d G . If dim A N ( G , d G ) > 0 , is dim A N ( G , d G ) infinite? (Brodskiy et al., preprint [6, Problem 5.3]). The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad–Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be groups with proper left invariant metrics. The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad–Nagata dimension n. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad–Nagata dimension is larger but finite.

Dimensions of locally and asymptotically self-similar spaces

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdimG = dim@1G + 1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X ×Y ) < asdimX+asdimY . Other applications are also given.

Systoles of Nilmanifolds with the Carnot―Caratheodory Metrics

We describe systolic constants for some important examples of metric spaces with an inner metric. Bibliography: 14 titles.

More examples of metric spaces where the inductive dimensions disagree

We give three examples of metric spaces where the inductive dimensions disagree. The two main examples are both N-compact. The first has closed sets which are not clopen Borel (Borel in the σ-algebra generated by the clopen sets). The second has weight ω 1 and, assuming all sets of cardinality ω 1 in the interval are Q-sets, contrasts the first by having all closed sets clopen Borel. The third example provides, for each α with ω 1⩽α⩽ c , a metric space of weight α with noncoinciding dimensions for which all subsets of weight less than α are strongly zero-dimensional. Each example answers a question posed by Mrowka.