Searcharxiv⌕ Search

arXiv subjects

Mikhail I. Ostrovskii

Publications and source records attributed to Mikhail I. Ostrovskii.

At least 37 records · Page 2Linked to original sources
arXiv2015

Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G. Schechtman.

math.FA↗
arXiv2014

Metric dimensions of minor excluded graphs and minor exclusion in groups

An infinite graph G is minor excluded if there is a finite graph that is not a minor of G. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

math.GR↗
arXiv2014

Connections between metric characterizations of superreflexivity and the Radon-Nikodým property for dual Banach spaces

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample.

math.FA↗
arXiv2013

Radon-Nikodým property and thick families of geodesics

Banach spaces without the Radon-Nikodým property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family $T$ of geodesics joining points $u$ and $v$ in a metric space is called {\it thick} if there is $α>0$ such that for every $g\in T$ and for any finite collection of points $r_1,...,r_n$ in the image of $g$, there is another $uv$-geodesic $\widetilde g\in T$ satisfying the conditions: $\widetilde g$ also passes through $r_1,...,r_n$, and, possibly, has some more common points with $g$. On the other hand, there is a finite collection of common points of $g$ and $\widetilde g$ which contains $r_1,...,r_n$ and is such that the sum of maximal deviations of the geodesics between these common points is at least $α$.

math.FA↗
arXiv2013

On metric characterizations of the Radon-Nikodým and related properties of Banach spaces

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond. The paper also contains related characterizations of reflexivity and the infinite tree property.

math.FA↗
arXiv2011

Test-space characterizations of some classes of Banach spaces

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_α\}_{α\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_α\}_{α\in A}$ admit uniformly bilipschitz embeddings into $X$. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M.I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence $\{X_m\}_{m=1}^\infty$ of finite-dimensional Banach spaces there is a sequence $\{H_n\}_{n=1}^\infty$ of finite connected unweighted graphs with maximum degree 3 such that the following conditions on a Banach space $Y$ are equivalent: (A) $Y$ admits uniformly isomorphic embeddings of $\{X_m\}_{m=1}^\infty$; (B) $Y$ admits uniformly bilipschitz embeddings of $\{H_n\}_{n=1}^\infty$. The second part of the paper is devoted to the case when $\{X_m\}_{m=1}^\infty$ is an increasing sequence of spaces. It is shown that in this case the class of spaces given by (A) can be characterized using one test-space, which can be chosen to be an infinite graph with maximum degree 3.

math.FA↗
arXiv2011

Different forms of metric characterizations of classes of Banach spaces

For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits uniformly bilipschitz embeddings of elements of the sequence H.

math.FA↗
arXiv2011

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 $\ell_1$ with uniformly bounded distortions. This result answers the problem posed by N. Linial, A. Magen, and A. Naor (2002).

math.MG↗
arXiv2011

Auerbach bases and minimal volume sufficient enlargements

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement which is a parallelepiped, some spaces have "exotic" minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having "exotic" minimal-volume sufficient enlargements in terms of Auerbach bases.

math.FA↗
arXiv2011

Embeddability of locally finite metric spaces into Banach spaces is finitely determined

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in the proof is: each locally finite subset of an ultraproduct $X^\mathcal{U}$ admits a bilipschitz embedding into $X$. We explain how this result can be used to prove analogues of the main results for other classes of embeddings.

math.FA↗
arXiv2010

Weak$^*$ closures and derived sets in dual Banach spaces

The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^*$ be an absolutely convex subset. Then $A^{(1)}={\bar{A}\,}^*$.

math.FA↗
arXiv1997

Distances between Banach spaces

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces $X$ and $Y$, the Kadets distance is defined to be the infimum of the Hausdorff distance $d(B_X,B_Y)$ between the respective closed unit balls over all isometric linear embeddings of $X$ and $Y$ into a common Banach space $Z.$ This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of $d(B_X,B_Y)$ over all isometric embeddings into a common metric space $Z$. We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has applications to the theory of complex interpolation.

math.FA↗
arXiv1994

Subspaces containing biorthogonal functionals of bases of different types

The paper is devoted to two particular cases of the following general problem. Let $α$ and $β$ be two types of bases in Banach spaces. Let a Banach space $X$ has bases of both types and a subspace $M\subset X^*$ contains the sequence of biorthogonal functionals of some $α$-basis in $X$. Does $M$ contain a sequence of biorthogonal functionals of some $β$-basis in $X$? The following particular cases are considered: $(α, β)$=(Schauder bases, unconditional bases), $(α, β)$=(Nonlinear operational bases, linear operational bases). The paper contains an investigation of some of the spaces constructed by S.Belle\-not in ``The $J$-sum of Banach spaces'', J. Funct. Anal. {\bf 48} (1982), 95--106. (These spaces are used in some examples.)

math.FA↗
arXiv1994

Classes of Banach spaces stable and unstable with respect to the opening

The paper is a complement to the survey: M.I.Ostrovskii "To\-po\-lo\-gies on the set of all subspaces of a Banach space and related questions of Banach space geometry", Quaestiones Math. (to appear). It contains proofs of some results on stability of properties of Banach spaces with respect to the geometric opening stated in the survey without proofs. Some results of the present paper are of independent interest, in particular the description of a predual property of the Banach--Saks property.

math.FA↗
arXiv1994

Quojections without Banach subspaces

A quojection (projective limit of Banach spaces with surjective linking mappings) without infinite dimensional Banach subspaces is constructed. This results answers a question posed by G.Metafune and V.B.Moscatelli.

math.FA↗
arXiv1993

On complemented subspaces of sums and products of Banach spaces

It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces. (This is a revised version of the paper by the same name formerly contained in the file ostrovskiicompsbsp.atx.)

math.FA↗
arXiv1993

Structure of total subspaces of dual Banach spaces

Let $X$ be a separable nonquasireflexive Banach space. Let $Y$ be a Banach space isomorphic to a subspace of $X^*$. The paper is devoted to the following questions: 1. Under what conditions does there exist an isomorphic embedding $T:Y\to X^*$ such that subspace $T(Y)\subset X^*$ is total? 2. If such embeddings exist, what are the possible orders of $T(Y)$? Here we need to recall some definitions. For a subset $M\subset X^*$ we denote the set of all limits of weak$^*$ convergent sequences in $M$ by $M_{(1)}$. Inductively, for ordinal number $α$ we let $$M_{(α)}=\cup_{β<α}(M_{(β)})_{(1)}.$$ The least ordinal $α$ for which $M_{(α)}= M_{(α+1)}$ is called the {\it order} of $M$.

math.FA↗