We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal.
In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give precise constructions of convex normal tilings of the unit sphere, the unit ball or in general of any convex body.
In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a ball of radius r and is contained in a ball of radius R.
In this paper, we define for the first time the local Morrey-type space associated with ball quasi-Banach function spaces and show the related series of properties. In addition, Hardy-Littlewood maximal operator's boundedness is proved. We investigate nonsmooth decomposition of the local Morrey-type space associated with ball quasi-Banach function spaces via the Hardy local Morrey-type spaces associated with ball quasi-Banach function spaces. And we consider Hardy operator's boundedness.
A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball, answering in this way a …
In this work, we introduce the notion of Ball dentable property in Banach spaces. We study certain stability results for the $w^*$-Ball dentable property leading to a discussion on Ball dentability in the context of ideals of Banach spaces. We prove that the $w^*$-Ball-dentable property can be lifted from an $M$-ideal to the whole Banach Space. We also prove similar results for strict ideals of a Banach space. We note that the space $C(K,X)^*$ has $w*$-Ball dentable property when $K$ is dispersed and $X^\ast$ has the $w^*$-Ball dentable property.
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry.
We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).
Let $E$ be a $(\mathrm{IV})$-polyhedral Banach space. We show that, for each $ε>0$, $E$ admits an $ε$-equivalent $\mathrm{(V)}$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points. In particular, we obtain that every separable isomorphically polyhedral Banach space, for each $ε>0$, admits an $ε$-equivalent $(\mathrm{V})$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points.
We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the $1$-sum of two Banach spaces does not have property (1), but it has property (2) if both the spaces have property (1); the Banach space $C[0,1]$ does not have property (2), although it has property (3).
We study the behaviour of Whitley's thickness constant of a Banach space with respect to $\ell_p$-products and we compute it for classical $L_p$-spaces.
A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\ell^p$-sums for $1<p<\infty$. We will show here that for any absolute, normalised norm $\|\cdot\|_E$ on $\mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $X\oplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $\|\cdot\|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.
It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E \to B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_1$ then every non-expansive bijection $F: B_E \to B_M$ is an isometry.
We study biorthogonal sequences with special properties, such as weak or weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig theorem. This result is applied to embed analytic disks in the fiber over 0 of the spectrum of H^infinity (B), the algebra of bounded analytic functions on the unit ball B of an arbitrary infinite dimensional Banach space. Various other embedding theorems are obtained. For instance, if the Banach space is superreflexive, then the unit ball of a Hilbert space of uncountable dimension can be embedded analytically in the fiber over 0 via an embed…
In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal $κ$, a Banach space $X$ contains an isomorphic copy of $\ell_…
We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every slice of its unit ball has diameter 2 and satisfying that its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small, which answers an open problem and stresses the differences of diameter between slices and relatively weakly open subsets of the unit ball in Banach spaces.
The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions $f$ on the unit ball $B_E$ of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of $(1-\|x\|^2) \|f'(x)\|$ on $B_E$ and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms $ϕ$ of $B_E$. We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on $B_{E}$ are Bloch functions in both ways.
We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hau…
We study convexity and starlikeness of quasihyperbolic and distance ratio metric balls on Banach spaces. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.
We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and, however, its unit ball still contains convex combinations of slices with diameter arbitrarily small, which improves in a optimal way the known results about the size of this kind of subsets in Banach spaces.
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a $1$-unconditional basis. A norm one element $x$ in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance $2$ from $x$. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an uncon…
A Banach space has the ball huskable property ($BHP$) if the closed unit ball has weakly open sets of arbitrarily small diameter. We can analogously define $w^*$-$BHP$ in the dual space. In this short note, we study these properties in the context of ideals in Banach spaces. The notion of an ideal, was introduced by Godefroy, Kalton and Saphar. We show that if a Banach space $X$ has $BHP,$ then any $M$-ideal of $X$ also has $BHP.$ We further show that if $Y$ is an $M$-ideal of $X,$ then $Y^*$ has $w^*$-$BHP$ implies $X^*$ has $w^{*}$-$BHP.$ We use this result to prove that for a compact Hausdo…
A Daugavet-point (resp.~$Δ$-point) of a Banach space is a norm one element $x$ for which every point in the unit ball (resp.~element $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from $x$. A Banach space has the well-known Daugavet property (resp.~diametral local diameter 2 property) if and only if every norm one element is a Daugavet-point (resp.~$Δ$-point). This paper complements the article "Delta- and Daugavet-points in Banach spaces" by T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk, where the study of the existence of Daugavet- and $Δ$-po…
We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter 2, which answer an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combinat…
We introduce extensions of $Δ$-points and Daugavet points in which slices are replaced by relative weakly open subsets (super $Δ$-points and super Daugavet points) or by convex combinations of slices (ccs $Δ$-points and ccs Daugavet points). We first give a general overview on these new concepts and provide some isometric consequences on the spaces. As examples: if a Banach space contains a super $Δ$-point, then it does not admit an unconditional FDD with suppression constant smaller than two; if a real Banach space contains a ccs $Δ$-point, then it does not admit a one-unconditional basis; if…
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $π$-property. Besid…
In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in \cite{B3}, \cite{BR} ,\cite{EW}, \cite{GM} was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in \cite{GGMS}. We introduce the the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, $…
We study Banach spaces with a weak stable unit ball, that is Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the class of $L_1$ preduals with a weak stable unit ball agree with those $L_1$ preduals which are purely atomic, that is preduals of $\ell_1(Γ)$ for some set $Γ$, getting in this way a complete geometrical characterization of purely atomic preduals of $L_1$, which answers a setting problem. As a consequence, we prove the equivalence for $L_1$ preduals of differen…
For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphisms $\mathcal M(\mathcal A)$ on $\mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0.$
The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related geometric properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrar…
We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.
Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$ and let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contr…
We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows by our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction proving existence of a star-finite covering of $c_0(Γ)$ by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countabl…
A $Δ$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if and only if every norm one element is a Daugavet-point. We show that $Δ$- and Daugavet-points are the same in $L_1$-spaces, $L_1$-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the uni…
We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. These ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids with a given functional of the dual space. The term asymptotical refers to the fact that we take '$\limsup$' over finite-dimensional subspaces. This leads to some isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem…
the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm.
