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We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L 1-embeddable metric spaces also embed in L 1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into Lp , 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in Lp . This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to Lp -spaces, 1 ≤ p < ∞.

The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of $\ell_1$ spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal roadmaps, are characterized. Possible obstacles for a TC space on a finite metric space $X$ preventing them from containing subspaces isometric to $\ell_\infty^n$ have been found in terms of the canonical graph of $X$. The fact that TC spaces on diamond graphs do not contain $\ell_\infty^4$ isometrically has been derived. In addition, a short overview of known results on the isometric structure of TC spaces on finite metric spaces is presented.

This article is a continuation of our article in Dilworth et al. (Can J Math 72:774–804, 2020). We construct orthogonal bases of the cycle and cut spaces of the Laakso graph $$\mathcal {L}_n$$ . They are used to analyze projections from the edge space onto the cycle space and to obtain reasonably sharp estimates of the projection constant of $${\text {Lip}}_0(\mathcal {L}_n)$$ , the space of Lipschitz functions on $$\mathcal {L}_n$$ . We deduce that the Banach–Mazur distance from $${\mathrm{TC}}\quad (\mathcal {L}_n)$$ , the transportation cost space of $$\mathcal {L}_n$$ , to $$\ell _1^N$$ of the same dimension is at least $$(3n-5)/8$$ , which is the analogue of a result from [op. cit.] for the diamond graph $$D_n$$ . We calculate the exact projection constants of $${\text {Lip}}_0(D_{n,k})$$ , where $$D_{n,k}$$ is the diamond graph of branching k. We also provide simple examples of finite metric spaces, transportation cost spaces on which contain $$\ell _\infty ^3$$ and $$\ell _\infty ^4$$ isometrically.

We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.

We prove that finite lamplighter groups { Z 2 ≀ Z n } n ≥ 2 \{\mathbb {Z}_2\wr \mathbb {Z}_n\}_{n\ge 2} with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space and therefore form a set of test spaces for superreflexivity. Our proof is inspired by the well-known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover Z 2 ≀ Z n \mathbb {Z}_2\wr \mathbb {Z}_n by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.

It is known that if finite subsets of a locally finite metric space M admit C-bilipschitz embeddings into l p ( 1 ≤ p ≤ ∞ ) , then for every e > 0 , the space M admits a ( C + e ) -bilipschitz embedding into l p . The goal of this paper is to show that for p ≠ 2 , ∞ this result is sharp in the sense that e cannot be dropped out of its statement.

Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces.

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177–194, 2008). Transportation cost spaces are also known as Arens–Eells, Lipschitz-free, or Wasserstein 1 spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete metric spaces such that transportation cost spaces on them do not contain isometric copies of $$\ell _1$$, this result answers a question raised by Cuth and Johanis (Proc Am Math Soc 145(8):3409–3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of $$\ell _1$$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to $$\ell _\infty ^d$$ of the corresponding dimension, and that for all finite metric spaces M, except a very special class, the infimum of all seminorms for which the embedding of M into the corresponding seminormed space is isometric, is not a seminorm.

The present paper deals with some structural properties of transportation cost spaces, also known as Arens-Eells spaces, Lipschitz-free spaces and Wasserstein spaces. The main results of this work are: (1) A necessary and sufficient condition on an infinite metric space $M$, under which the transportation cost space on $M$ contains an isometric copy of $\ell_1$. The obtained condition is applied to answer the open questions asked by C\'uth and Johanis (2017) concerning several specific metric spaces. (2) The description of the transportation cost space of a weighted finite graph $G$ as the quotient $\ell_1(E(G))/Z(G)$, where $E(G)$ is the edge set and $Z(G)$ is the cycle space of $G$. This is a generalization of the previously known result to the case of any finite metric space.

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., 37 pages, 5 figures, some small improvements of presentation

The main result: for every sequence $\{\omega_m\}_{m=1}^\infty$ of positive numbers ($\omega_m>0)$ there exists an isometric embedding $F:[0,1]\to L_1[0,1]$ which is nowhere differentiable, but for each $t\in [0,1]$ the image $F_t$ is infinitely differentiable on $[0,1]$ with bounds $\max_{x\in[0,1]}|F_t^{(m)}(x)|\le\omega_m$ and has an analytic extension to the complex plane which is an entire function.

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.

We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property does not necessarily contain a bilipschitz image of a thick family of geodesics. This is done by showing that any thick family of geodesics is not Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikod\'ym property contains a bilipschitz image of a thick family 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 thick if there is α > 0 such that for every g ∈ T and for any finite collection of points r 1 , … , r n in the image of g , there is another u v -geodesic g ˜ ∈ T satisfying the conditions: 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 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 .

This paper is devoted to estimates of the spanning tree congestion for some planar graphs. We present three main results: (1) We almost determined (up to [math] ) the maximal possible spanning tree congestion for planar graphs. (2) The value of congestion indicator introduced by Ostrovskii [Discrete Math. 310, 1204–1209] can be very far from the value of the spanning tree congestion. (3) We find some more examples in which the congestion indicator can be used to find the exact value of the spanning tree congestion.

The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is based on the theory of equal-signs-additive sequences developed by Brunel and Sucheston (1975-1976). We also show that some of the low-distortion embeddability results obtained using this method cannot be obtained using the method based on the factorization between the summing basis and the unit vector basis of $\ell_1$, which was used by Bourgain (1986) and Johnson and Schechtman (2009)., Many changes, more details of proofs, new figures

The main purpose of the paper is to present some recent results on metric characterizations of superreflexivity and the Radon–Nikodým property.

Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other, for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. This question was answered in the negative by Ostrovskii (2014), who left it open to determine the order of growth of the distortions. The main purpose of this paper is to get a sharp-up-to-a-logarithmic-factor estimate for the distortions of embeddings of binary trees into diamond graphs, and, more generally, into diamond graphs of any finite branching $k\ge 2$. Estimates for distortions of embeddings of diamonds into infinitely branching diamonds are also obtained., Improved presentation and corrected references

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$ itself admits a bilipschitz embedding into $X$ with distortion $\le \alpha\cdot C$; (2) $D(X)=\alpha^+$ if, for every $\varepsilon>0$, the condition $D(X)\le\alpha+\varepsilon$ holds, while $D(X)\le\alpha$ does not; (3) $D(X)\le \alpha^+$ if $D(X)=\alpha^+$ or $D(X)\le \alpha$. It is known that $D(X)$ is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) $D((\oplus_{n=1}^\infty X_n)_p)\le 1^+$ for every nested family of finite-dimensional Banach spaces $\{X_n\}_{n=1}^\infty$ and every $1\le p\le \infty$. (2) $D((\oplus_{n=1}^\infty \ell^\infty_n)_p)=1^+$ for $1, Comment: This version is a significant update of version 1 since the main result of version 1 turned out to be known (Kalton-Lancien 2008)

The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3/2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n 3/2 .

We consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

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 $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.

We study properties of solution sets of inequalities of the form $$X^* AX + B^* X + X^* B + C \leqslant 0,$$ , where A, B, and C are bounded Hilbert space operators and A and C are self-adjoint. The following properties are considered: closedness and inferior points in Standard operator topologies, convexity, and nonemptiness.

Properties of sets of solutions to inequalities of the form $$X^{*} AX + B^{*}X + X^{*}B + C \leq 0$$ are studied, where A, B, C are bounded Hilbert space operators, A and C are self-adjoint. Properties under consideration: closeness and interior points in standard operator topologies, convexity, non-emptiness.

In this paper we study linear fractional relations defined in the following way. Let${\cal B}$i,${\cal B}$'i,i= 1,2, be Banach spaces. We denote the space of bounded linear operators by${\cal L}$. LetTε${\cal L}$(${\cal B}$1⊕${\cal B}$2,${\cal B}$'1⊕${\cal B}$'2). To each such operator there corresponds a 2 × 2 operator matrix of the form(*)whereTijε${\cal L}$(${\cal B}$j,${\cal B}$'i. For each suchTwe define a set-valued mapGTfrom${\cal L}$(${\cal B}$1,${\cal B}$2) into the set of closed affine subspaces of${\cal L}$(${\cal B}$'1,${\cal B}$'2) byThe mapGTis called alinear fractional relation.The paper is devoted to the following two problems.•Characterization of operator matrices of the form (*) for which the setGT(K) is non-empty for eachKin some open ball of the space${\cal L}$(${\cal B}$1,${\cal B}$2).•Characterizations of quadruples (${\cal B}$1,${\cal B}$2,${\cal B}$'1,${\cal B}$'2) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem “a bounded entire function is constant”.

Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. The well-known fact that Laakso graphs are uniformly doubling but diamond graphs are not, immediately implies that diamond graphs do not admit uniformly bilipschitz embeddings into Laakso graphs. The main goal of this paper is to prove that Laakso graphs do not admit uniformly bilipschitz embeddings into diamond graphs.

In this paper we study linear fractional relations defined in the following way. Let ℋi and ℋi′, i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from ℋj to ℋi′ by L (ℋj, ℋi′). Let T ∈ ℒ(ℋ1 ⊕ ℋ2, ℋ1′ ⊕ ℋ2′). To each such operator there corresponds a 2 × 2 operator matrix of the form where Tij ∈ ℒ (ℋj, ℋi′), i, j = 1, 2. For each such T we define a set-valued map GT from ℒ(ℋ1, ℋ2) into the set of closed affine subspaces of ℒ(ℋ1′, ℋ2′) by GT (K ) = {K′ ∈ ℒ(ℋ1′, ℋ2′) : T21 + T22K = K′(T11 + T12K )} . The map GT is called a linear fractional relation . The main result of the paper is the description of operator matrices of the form (.) for which the relation GT is defined on some open ball of the space ℒ(ℋ1, ℋ2). Linear fractional relations are natural generalizations of linear fractional transformations studied by M. G. Krein and Yu. L. Smuljan (1967). The study of both linear fractional transformations and linear fractional relations is motivated by the theory of spaces with an indefinite metric and its applications. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Consider a bounded linear operator T between Banach spaces ℬ, ℬ′ which can be decomposed into direct sums ℬ = ℬ1 ⌖ ℬ2, ℬ′ = ℬ1′ ⌖ ℬ2′. Such linear operator can be represented by a 2 × 2 operator matrix of the form where Tij ∈ ℒ(ℬj, ℒi′) i, j = 1, 2. (By ℒ(ℬj, ℒi′) we denote the space of bounded linear operators acting from ℬj to ℬi′ (i, j = 1, 2).) The map GT from L (B1, B2) into the set of closed affine subspaces of ℒ(ℬ1′ ℬ2′), defined by is called a linear fractional relation associated with T.Such relations can be considered as a generalisation of linear fractional transformations which were studied by many authors and found many applications. Many traditional and recently discovered areas of application of linear fractional transformations would benefit from a better understanding of the behaviour of linear fractional relations. The present paper is devoted to analogues of the Liouville theorem “a bounded entire function is constant” for linear fractional relations.

This paper is devoted to the study of operators satisfying the condition $$ ||A||\, = \max \{ \rho (AB):||B||\, = 1\} , $$ where ρ stands for the spectral radius; and Banach spaces in which all operators satisfy this condition. Such spaces are called V−spaces. The present paper contains partial solutions of some of the open problems posed in the first part of the paper. The main results: (1) Each subspace of l p (1

We denote by $B_Y$ the unit ball of a normed linear space $Y$ . Definition . A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed space $X$ is called a 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 $P(B_Y)\,{\subset}\,A$ . Minimal-volume sufficient enlargements are determined for two-dimensional spaces. The main results are: (i) Each minimal-volume sufficient enlargement for a two-dimensional space is a parallelogram or a hexagon. (ii) If a two-dimensional normed space $X$ has a minimal-volume sufficient enlargement that is not a parallelogram, then $B_X$ is linearly equivalent to the regular hexagon.

Let G be a graph and let T be a tree with the same vertex set. Let e be an edge of T and A e and B e be the vertex sets of the components of T obtained after removal of e . Let E G ( A e , B e ) be the set of edges of G with one endvertex in A e and one endvertex in B e . Let ec (G:T)≔ max e |E G (A e ,B e )|. The paper is devoted to minimization of ec( G : T ) • Over all trees with the same vertex set as G . • Over all spanning trees of G . These problems can be regarded as “congestion” problems.

One of the important problems of the local theory of Banach Spaces can be stated in the following way. We consider a condition on finite sets in normed spaces that makes sense for a finite set any cardinality. Suppose that the condition is such that to each set satisfying it there corresponds a constant describing “how well” the set satisfies the condition.The problem is:Suppose that a normed space X has a set of large cardinality satisfying the condition with “poor” constant. Does there exist in X a set of smaller cardinality satisfying the condition with a better constant?In the paper this problem is studied for conditions associated with one of R.C. James's characterisations of superreflexivity.

Among all linear projections onto a given linear subspace L in R n we select those that minimize the volume of the image of the cube {x:|xi|⩽1}. The paper is devoted to a description of the shape of such images of the cube. The shape is characterized in terms of zonotopes spanned by scalar multiples of rows of totally unimodular matrices.

In this paper we review ongoing work on operators having norm-preserving extensions to every overspace. We call them Hahn-Banach operators.

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. 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, like ℤ3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

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 setMwhose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold and thatM=l2is a counterexample.

We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.

We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces (X, Y ) such that there exists a Hahn-Banach operator T : X → Y of rank k. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman. Everywhere in this paper we consider only real linear spaces. Our starting point is the classical Hahn-Banach theorem ([H], [B1]). The form of the Hahn-Banach theorem we are interested in can be stated in the following way. Hahn-Banach Theorem. Let X and Y be Banach spaces, T : X → Y a bounded linear operator of rank 1 and Z a Banach space containing X as a subspace. Then there exists a bounded linear operator T : Z → Y satisfying (a) ||T || = ||T ||; (b) T x = Tx for every x ∈ X. Definition 1. An operator T : Z → Y satisfying (a) and (b) for a bounded linear operator T : X → Y is called a norm-preserving extension of T to Z. The Hahn-Banach theorem is one of the basic principles of linear analysis. It is quite natural that there exists a vast literature on generalizations of the HahnBanach theorem for operators of higher rank. See papers by G. P. Akilov [A], J. M. Borwein [Bor], B. L. Chalmers and B. Shekhtman [CS], G. Elliott and I. Halperin [EH], D. B. Goodner [Go], A. D. Ioffe [I], S. Kakutani [Kak], J. L. Kelley [Kel], J. Lindenstrauss [L1], [L2], L. Nachbin [N1] and M. I. Ostrovskii [O], representing different directions of such generalizations, and references therein. There exist two interesting surveys devoted to the Hahn-Banach theorem and its generalizations; see G. Buskes [Bus] and L. Nachbin [N2]. We shall use the following natural definition. Definition 2. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. Received by the editors February 9, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 47A20.

We study the shape of minimal-volume shadows of a cube in a given subspace. First we prove an essentially known result that for every subspace L the set of minimal-volume shadows in L contains a parallelepiped. Our main result is that for some subspaces there exist minimal-volume shadows that are far from parallelepipeds with respect to the Banach–Mazur distance.

D e f i n i t i o n . A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection \(P:Y\to X\) such that P(B(Y)) \(\subset\) A (by B we denote the unit ball). ¶The notion of sufficient enlargement is implicit in the paper: B. Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95, 451 - 465 (1960). It was explicitly introduced by the author in: M. I. Ostrovskii, Generalization of projection constants: sufficient enlargements, Extracta Math. 11, 466 - 474 (1996). ¶The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces.

Let p be a class of Banach spaces and let T = {Tα}αєA be a set of metric spaces. We say that T is a set of test-spaces for P if the following two conditions are equivalent: (1) X ∉ P; (2) The spaces {Tα}αєA admit uniformly bilipschitz embeddings into X.