Embeddings Of Metric Spaces Research Articles

Quadratic embedding constants of graphs: Bounds and distance spectra

The quadratic embedding constant (QEC) of a finite, simple, connected graph G is the maximum of the quadratic form of the distance matrix of G on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. Next, we identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph G with either a complete or star graph can be computed in terms of the QEC of G. As an application of this result, we provide new families of examples of graphs of QE class.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

About convex structures on metric spaces

"In the present paper we study the relationships between different concepts of convex structures in metric spaces that that are related to the works of K. Menger [Menger, K. Untersuchungen \""{u}ber allgemeine Metrik. {\it Math. Ann.} {\bf100} (1928), 75--163], H. Busemann [Busemann, H. {\it The geometry of geodesics}, Academic Press, 1955], I. N. Herstein; J. Milnor [Herstein, I. N.; Milnor, J. An axiomatic approach to measurable utility. {\it Econometrica} {\bf21} (1953), 291--297], E. Michael [Michael, E. Convex structures and continuous selections. {\it Canad. J. Math.} {\bf11} (1959), 556--575], A. Nijenhuis [Nijenhuis, A. A note on hyperconvexity in Riemannian manifolds. {\it Canad. J. Math.} {\bf11} (1959), 576--582.], W. Takahashi; T. Shimizu [Shimizu, T.; Takahashi, W. Fixed points of multivalued mappings in certain metric spaces. {\it Topol. Methods Nonlinear Anal.} {\bf8} (1996), no. 1, 197--203 and Takahashi, W. A convexity in metric space and nonexpansive mappings I. {\it Kodai Math. Sem. Rep.} {\bf 22} (1970), 142--149], M. Taskovi\'{c} [Taskovi\'{c}, M. General convex topological spaces and fixed points. {\it Math. Moravica} {\bf 1} (1997), 127--134], Yu. A. Aminov [Aminov, Yu. A. Two-Dimensional Surfaces in 3-Dimensional and 4-Dimensional Euclidean Spaces. Results and Unsolved Problems. {\it Ukr. Math. J.} {\bf 71} (2019), no. 1, 1--38.], H. V. Machado [Machado, H. V. A characterization of convex subsets of normed spaces.{\it Kodai Math. Sem. Rep.} {\bf25} (1973), 307--320], and many other papers. Some well known examples of concrete convex structures are reexamined and the possibilities of different embeddings of metric spaces with convex structures are also studied. Corollary \ref{C5.1} states that the Bolyai-Lobachevskii plane and the Bolyai-Lobachevskii half-plane are not isometrically embedding in some strictly convex normed space. A characteristic of the invariant metric generated by a norm is presented (Proposition \ref{P4.1}). "

A remark on quasimöbius and quasiconformal maps

The main purpose of this note is to show a certain connection between quasiconformal and quasimobius maps. We prove that quasimobius maps are the natural boundary maps for quasiconformal maps from the upper half plane onto itself as well as for quasiconformal maps from the unit disc onto itself. In the first case we can remove the rather unpleasant condition that infinity is fixed. We think that the most interesting aspect of our observations is that the results concerning the unit disk follow by very simple arguments from the corresponding results on half planes by the use of quasimobius maps. Similar results concerning the disk are earlier proved by Krzyz (Quasicircles and harmonic measure. Ann Acad Sci Fenn. 12:19–24, 1987) and Douady and Earle (Conformally natural extension of homeomorphisms of the circle. Acta Math 157:23–48, 1986). However, using properties of quasimobius maps proved by Vaisala (Quasimobius maps. J Anal Math 44:218–234, 1984), Tukia and Vaisala (Quasisymmetric embeddings of metric spaces. Ann Acad Sci Fenn Ser A I Math 5:97–114, 1980) and Heinonen (Lectures on analysis on metric spaces. Berlin: Springer, 2001) we are able to prove some of these results in a much simpler way. In this respect our approach to these problems is more natural than the approach in Douady and Earle (Conformally natural extension of homeomorphisms of the circle. Acta Math 157:23–48, 1986) and Krzyz (Quasicircles and harmonic measure. Ann Acad Sci Fenn. 12:19–24, 1987). [See also Pommerenke (Boundary behaviour of conformal maps. Berlin: Springer, 1992)]. To the best of our knowledge the situation for the quasimobius maps is not considered in the literature. In a paper by Bonk et al. (Rigidity of Schottky sets. Am J Math 131(2):409–443, 2009) related results in higher dimensions are proved in Section 4.

Local Embeddings of Metric Spaces

In many application areas, complex data sets are often represented by some metric space and metric embedding is used to provide a more structured representation of the data. In many of these applications much greater emphasis is put on preserving the local structure of the original space than on maintaining its complete structure. This is also the case in some networking applications where small world phenomena in communication patterns have been observed. Practical study of embedding has indeed involved with finding embeddings with this property. In this paper we initiate the study of local embeddings of metric spaces and provide embeddings with distortion depending solely on the local structure of the space.

Distortion is Fixed Parameter Tractable

We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let M = M ( G ) be the shortest path metric of an edge-weighted graph G , with the vertex set V ( G ) and the edge set E ( G ), on n vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric M and integer d either constructs an embedding of M into the line with distortion at most d , or concludes that no such embedding exists. Our algorithm requires O ( nd 4 (2 d + 1) 2d ) time which is a significant improvement over the best previous algorithm that runs in time O ( n 4d+2 d O(1) ). Because of its apparent similarity to the notoriously hard Bandwidth Minimization problem, we find it surprising that this problem turns out to be fixed parameter tractable. We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small-distortion embeddings of weighted graph metrics. The running time of our algorithm is O ( n ( dW ) 4 (2 d + 1) 2dW ), where W is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric M ( G ) with maximum weight W < | V ( G )| can be embedded into the line with distortion at most d is NP-complete for every fixed rational d ≥ 2. This rules out any possibility of an algorithm with running time O (( nW ) h(d) ) where h is a function of d alone. Second, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree T with maximum degree Δ , embedding M into a shortest path metric of T is fixed parameter tractable, parameterized by ( Δ , d ).

Isometric embeddings of pretangent spaces in $ E^n$

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

The Nonlinear Geometry of Banach Spaces

We survey some of the recent developments in the nonlinear theory of Banach spaces, with emphasis on problems of Lipschitz and uniform homeomorphism and uniform and coarse embeddings of metric spaces.

Asymptotic dimension and uniform embeddings

We study uniform embeddings of metric spaces, which satisfy some asymptotic tameness conditions such as finite asymptotic dimension, finite Assouad–Nagata dimension, polynomial dimension growth or polynomial growth, into function spaces. We show how the type function of a space with finite asymptotic dimension estimates its Hilbert (or any ℓ^p -) compression. In particular, we show that the spaces of finite asymptotic dimension with linear type (spaces with finite Assouad–Nagata dimension) have compression rate equal to one. We show, without the extra assumption that the space has the doubling property (finite Assouad dimension), that a space with polynomial growth has polynomial dimension growth and compression rate equal to one. The method used allows us to obtain a lower bound on the compression of the lamplighter group ℤ ≀ ℤ , which has infinite asymptotic dimension.

Linearly rigid metric spaces and the embedding problem

We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. Various properties of linearly rigid spaces and related spaces are considered.

Remarks on coarse embeddings of metric spaces into uniformly convex Banach spaces

We study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex Banach space. By using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex Banach space is determined by its finite subspaces.

Embedding Levy families into Banach spaces

We prove that if a metric probability space with a usual concentration property embeds into a finite dimensional Banach space X, then X has a Euclidean subspace of a proportional dimension. In particular this yields a new characterization of weak cotype 2. We also find optimal lower estimates on embeddings of metric spaces with concentration properties into $ l^k_\infty $ , generalizing estimates of Bourgain—Lindenstrauss—Milman, Carl—Pajor and Gluskin.

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.

Embedding into the rectilinear grid

Abstract: We show that the embedding of metric spaces into the l 1 -grid Z 2 can be characterized inessentially the same fashion as in the case of the l 1 -plane R 2 . In particular, a metric space can beembedded into Z 2 iff every subspace with at most 6 points is embeddable. Moreover, if such an embed-ding exists, it can be constructed in polynomial time (for finite spaces). q 1998 John Wiley & Sons, Inc.Networks 32: 127–132, 1998 The rectilinear metric (alias l 1 -metric) is probably the for the embedding in the gridZ 2 (‘‘digital plane’’) en-simplest distance measure in R n . This explains why in dowed with the rectilinear distance (alias city block met-many cases rectilinear spaces are selected as host spaces ric) (see Fig. 1). Again, as in [1],wecantake advantagefor embedding a given metric space; we refer to Hubert of such notions as ‘‘d-split’’ and ‘‘totally decomposableet al. [7] for an application to multidimensional scaling metrics,’’ introduced and investigated in the case of finitewhere one aims at producing a visual display of a given metric spaces by Bandelt and Dress [2].data set and to Deza and Laurent [6] for other applica- Let (X, d) be a metric space. We will say that (X, d)tions. Surprisingly, the problem to characterize the metric satisfies theparity condition if d(u, £) /d(£, w) /d(w,subspaces of the rectilinear space of a given dimension u) is an even integer for any u, £, w ˆX (cf. [10] forseems to be much more difficult than is the analogous a straightforward metric characterization of bipartiteproblem for Euclidean spaces, where suitable criteria are graphs). To every pair S †{A, B} of nonempty subsetsgiven by the classical results of Menger and Schoenberg of X we associate the isolation index a

Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We construct a sequence of metric spaces (Mn) with cardMn=3n satisfying that for everyc 0 such that, if the Lipschitz distance fromMn to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].