Related papers: The lattice Sch\"affer constant

We obtain several new results on the simultaneous packing and covering constant $\gamma(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $\gamma^*(\mathcal{X})$. These constants measure how efficient a (lattice)…

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 $A$…

Recent results of A. Lerner concerning certain properties of the Fefferman-Stein maximal function are applied to show that $(\BMO, X)_\theta = X^\theta$, $0 < \theta < 1$, for a Banach lattice $X$ of measurable functions on $\mathbb R^n$…

In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $\varphi\in X^*\setminus \{0\}$ and $\psi:X\to {\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $\varphi\|_X^{*}$. Then, we…

The general problem we address is to develop new methods in the study of projection constants of Banach spaces of multivariate polynomials. The relative projection constant $\boldsymbol{\lambda}(X,Y)$ of a subspace $X$ of a Banach $Y$ is…

A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x \in X$ if $\lvert x_\alpha - x\rvert \wedge u$ converges to $0$ in order for all $u\in X_+$. This convergence has been investigated and applied in several recent…

For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $\beta(X,m)$ to be the infimum of $\delta\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets…

A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $\nu\colon\mathcal F\to X$ satisfying…

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the…

A net $(x_\alpha)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that…

A net $(x_\alpha)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_\alpha-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study…

Let $L$ be a sectorial operator of type $\alpha$ ($0 \leq \alpha < \pi/2$) on $L^2(\mathbb{R}^d)$ with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying certain size and regularity conditions. Define $$ S_{q,L}(f)(x) =…

A well-known theorem due to R. C. James states that a Banach space is reflexive if and only if every bounded linear functional attains its norm. In this note we study Banach lattices on which every (real-valued) lattice homomorphism attains…

In this paper, we consider a condition on subspaces in order to improve bounds given in the Bernstein's Lethargy Theorem (BLT) for Banach spaces. Let $d_1 \geq d_2 \geq \dots d_n \geq \dots > 0$ be an infinite sequence of numbers converging…

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ 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)…

In this note we find $\lambda>1$ and give an explicit construction of a separable Banach space $X$ such that there is no $\lambda$-Lipschitz retraction from $X$ onto any compact convex subset of $X$ whose closed linear span is $X$. This is…

Let $X$ and $E$ be $f$-algebras and $p:X \to E_+$ be a monotone vector norm. Then the triple $(X,p,E)$ is called a lattice-normed $f$-algebraic space. In this paper, we show a generalization of the extension of the Hahn-Banach theorem for…

We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family $\mathcal{S}_\alpha$, $\alpha<\omega_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

In this paper, we primarily focus on analyzing the stability property of phase retrieval by examining the bi-Lipschitz property of the map $\Phi_{\boldsymbol{A}}(\boldsymbol{x})=|\boldsymbol{A}\boldsymbol{x}|\in \mathbb{R}_+^m$, where…