We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.

We prove an extension of Pisier’s inequality (1986) with a dimension-independent constant for vector-valued functions whose target spaces satisfy a relaxation of the UMD property.

Main results of the paper are as follows:

(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.

(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.

Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

A precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X,Y)$ has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X,Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X,Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_{p}(\unicode[STIX]{x1D707})$ spaces fail to have this property when $p>2$. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial finitely branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a non-trivial $\aleph _{0}$-branching bundle graph. For the specific case of $L_{1}$, it is shown that every countably branching bundle graph bi-Lipschitzly embeds into $L_{1}$ with distortion no worse than $2$.

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.

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$

$n$

$d_{N}$

$N\times N$

$N$

$N$

$N=1$

$n$

$N\times N$

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 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 that 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.

This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.

We study the structure of the sets of symmetric sequences and spreading models of an Orlicz sequence space equipped with partial order with respect to domination of bases. In the cases that these sets are “small”, some descriptions of the structure of these posets are obtained.

The paper considers three questions about independent random points uniformly distributed in isotropic symmetric convex bodies $K, T_1,\ldots, T_s$. (a) Let $\varepsilon\,{\in}\, (0,1)$ and let $x_1,\ldots, x_N$ be chosen from K. Is it true that if $N\,{\geq}\, C(\varepsilon )n\log n$, then \[\left\| I-\frac{1}{NL_K^2}\sum_{i=1}^Nx_i\otimes x_i\right\|<\varepsilon\] with probability greater than $1\,{-}\,\varepsilon $? (b) Let $x_i$ be chosen from $T_i$. Is it true that the unconditional norm \[\|{\bf t}\|=\int_{T_1}\!{\ldots}\int_{T_s}\left\|\sum_{i=1}^st_ix_i\right\|_K\,dx_s\ldots dx_1\] is well comparable to the Euclidean norm in ${\mathbb R}^s$? (c) Let $x_1,\ldots, x_N$ be chosen from K. Let ${\mathbb E}\,(K,N):={\mathbb E}\,|{\rm conv}\{ x_1,\ldots, x_N\}|^{1/n}$ be the expected volume radius of their convex hull. Is it true that ${\mathbb E}\,(K,N)\,{\simeq}\, {\mathbb E}\,(B(n),N)$ for all N, where $B(n)$ is the Euclidean ball of volume 1?

It is proved that the answers to these questions are affirmative if there is a restriction to the class of unconditional convex bodies. The main tools come from recent work of Bobkov and Nazarov. Some observations about the general case are also included.

We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from $[\text{LMS}]$, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “phase transition”.

We consider the analogue of the $\wedge (p)$―problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the “best possible” result that if $2\,<\,p\,<\,4$ then a random subset of $N$ Legendre polynomials of size ${{N}^{4/p-1}}$ spans an Hilbertian subspace. We also answer a question of König concerning the structure of the space of polynomials of degree $n$ in various weighted ${{L}_{p}}$-spaces.