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Vegard Lima

Publications and source records attributed to Vegard Lima.

At least 19 recordsLinked to original sources
arXiv2025

Duality of Lipschitz-free spaces over ultrametric spaces

We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point.

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arXiv2024

Strictly convex norms and the local diameter two property

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $c_0(\Gamma)$ possesses this property. In the opposite direction, we show that any Banach space which is strictly convex renormable and contains a complemented copy of $c_0(\mathbb N),$ admits an equivalent strictly convex norm for which the space has the local diameter two property. In particular, this enables us to construct a strictly convex norm on $c_0(\Gamma),$ where $\Gamma$ is uncountable, for which the space has a 1-unconditional basis and the local diameter two property.

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arXiv2024

Delta-points and their implications for the geometry of Banach spaces

We show that the Lipschitz-free space with the Radon--Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point. Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces.

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arXiv2023

A relative version of Daugavet-points and the Daugavet property

We introduce relative versions of Daugavet-points and the Daugavet property, where the Daugavet-behavior is localized inside of some supporting slice. These points present striking similarities with Daugavet-points, but lie strictly between the notions of Daugavet- and $Δ$-points. We provide a geometric condition that a space with the Radon--Nikodým property must satisfy in order to be able to contain a relative Daugavet-point. We study relative Daugavet-points in absolute sums of Banach spaces, and obtain positive stability results under local polyhedrality of the underlying absolute norm. We also get extreme differences between the relative Daugavet property, the Daugavet property, and the diametral local diameter 2 property. Finally, we study Daugavet- and $Δ$-points in subspaces of $L_1(μ)$-spaces. We show that the two notions coincide in the class of all Lipschitz-free spaces over subsets of $\mathbb{R}$-trees. We prove that the diametral local diameter 2 property and the Daugavet property coincide for arbitrary subspaces of $L_1(μ)$, and that reflexive subspaces of $L_1(μ)$ do not contain $Δ$-points. A subspace of $L_1[0,1]$ with a large subset of $Δ$-points, but with no relative Daugavet-point, is constructed.

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arXiv2022

Asymptotic geometry and delta-points

We study Daugavet- and $Δ$-points in Banach spaces. A norm one element $x$ is a Daugavet-point (respectively a $Δ$-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing $x$) you can find another element of distance as close to $2$ from $x$ as desired. In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $Δ$-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $Δ$-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally we prove that there exists a superreflexive Banach space with a Daugavet- or $Δ$-point provided there exists such a space satisfying a weaker condition.

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arXiv2021

Delta-points in Banach spaces generated by adequate families

We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by adequate families $\mathcal A$ where $1 \le p < \infty$. In the case the familiy $\mathcal A$ is regular and $p=1,$ these spaces are known as combinatorial Banach spaces. When $p > 1$ we prove that neither $h_{\mathcal{A},p}$ nor its dual contain delta-points. Under the extra assumption that $\mathcal A$ is regular, we prove that the same is true when $p=1.$ In particular the Schreier spaces and their duals fail to have delta-points. If $\mathcal A$ consists of finite sets only we are able to rule out the existence of delta-points in $h_{\mathcal{A},1}$ and Daugavet-points in its dual. We also show that if $h_{\mathcal{A},1}$ is polyhedral, then it is either (I)-polyhedral or (V)-polyhedral (in the sense of Fonf and Veselý).

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arXiv2020

Daugavet- and delta-points in Banach spaces with unconditional bases

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 unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than $2$. We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a $1$-unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a $1$-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

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arXiv2020

On Daugavet indices of thickness

Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the Daugavet index of thickness, which helps us solve an open problem which connect the Daugavet indices with the Daugavet equation. Moreover, we will improve the formerly known estimates of the behavior of Daugavet index on direct sums of Banach spaces by establishing sharp bounds. As a consequence of our results we prove that, for every $0<δ<2$, there exists a Banach space where the infimum of the diameter of convex combinations of slices of the unit ball is exactly $δ$, solving an open question from the literature. Finally, we prove that an open question posed by Ivakhno in 2006 about the relation between the radius and diameter of slices has a negative answer.

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arXiv2018

Delta- and Daugavet-points in Banach spaces

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 unit sphere are $Δ$-points, but where none of them are Daugavet-points. We also study the property that the unit ball is the closed convex hull of its $Δ$-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all $C(K)$ spaces, $K$ infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

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arXiv2018

Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open

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 Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces.

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arXiv2018

Symmetric strong diameter two property

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by $\ell_\infty$-sums, and working with weak star slices we show that $\text{Lip}_0(M)$ have the weak star version of the property for several classes of metric spaces $M$.

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arXiv2016

Octahedral norms in tensor products of Banach spaces

We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.

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arXiv2016

Diameter two properties, convexity and smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.

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arXiv2016

Almost square and octahedral norms in tensor products of Banach spaces

The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.

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arXiv2015

Almost square Banach spaces

We single out and study a natural class of Banach spaces -- almost square Banach spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i-y\|$ is almost one. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square. We show that every separable space containing a copy of $c_0$ can be renormed to be almost square. A local and a weak version of almost square spaces are also studied.

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arXiv2014

On thickness and thinness of Banach spaces

The aim of this note is to complement and extend some recent results on Whitley's indices of thinness and thickness in three main directions. Firstly, we investigate both the indices when forming $\ell_p$-sums of Banach spaces, and obtain formulas which show that they behave rather differently. Secondly, we consider the relation of the indices of the space and a subspace. Finally, every Banach space $X$ containing a copy of $c_0$ can be equivalently renormed so that in the new norm $c_0$ is an M-ideal in $X$ and both the thickness and thinness index of $X$ equal 1.

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arXiv2013

Almost isometric ideals in Banach spaces

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

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