arXiv subjects

Johann Langemets

Publications and source records attributed to Johann Langemets.

At least 19 recordsLinked to original sources

arXiv2026

Dual Banach spaces with the ball-covering property

We study ball-covering properties of dual Banach spaces and their connections with the geometry of predual unit balls. One of our main results shows that, for every separable Banach space $X$, the unit ball $B_X$ is a slicely countably determined set if and only if $\operatorname{bc}(X^*)=1$, where $\operatorname{bc}(\cdot)$ is the ball-covering index introduced by A. J. Guirao, A. Lissitsin, and V. Montesinos. We obtain several sufficient conditions for the uniform ball-covering property in dual spaces, including duals of spaces with a $K$-unconditional basis for $K<2$, and duals of separable spaces whose unit ball is the closed convex hull of a set of uniformly strongly exposed points. The constant $2$ is sharp: there is a space with a $2$-unconditional basis whose dual fails the ball-covering property. Applications are given to spaces of operators and to Lipschitz spaces. In particular, $\mathcal L(L_p[0,1])$ has the uniform ball-covering property for every $1<p<\infty$, which answers a question posed by Q. Bao, R. Liu, and J. Shen. As an application to Lipschitz spaces, we prove that $\operatorname{Lip}_0(M)$ has the uniform ball-covering property whenever $M$ is a separable complete ultrametric or Hölder metric space.

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arXiv2026

Transfinite Daugavet property

We extend the Daugavet property and a perfect version of it to transfinite cardinals in order to distinguish between spaces with the ordinary Daugavet property by some kind of complexity (topological, density\ldots), providing a number of examples and results. First, we characterise the transfinite Daugavet $C(K)$ spaces in terms of a cardinal index $\mathfrak r(K)$, which generalises the notion of the reaping number of a Boolean algebra. Besides, the perfect Daugavet property characterizes the absence of $G_δ$-points in $K$. We also study several inheritance results of the transfinite Daugavet properties by almost isometric ideals, absolute sums, and tensor product spaces, with a number of applications. We classify these properties for $L_1(μ)$ and $L_\infty(μ)$ spaces in terms of the Maharam's decomposition of the measure. We also show that the space of Lipschitz functions $\Lip(M)$ on a complete length metric space has the $ω$-perfect Daugavet property, improving the previous knowledge.

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arXiv2025

The super Alternative Daugavet property for Banach spaces

We introduce the super alternative Daugavet property (super ADP) which lies strictly between the Daugavet property and the Alternative Daugavet property as follows. A Banach space $X$ has the super ADP if for every element $x$ in the unit sphere and for every relatively weakly open subset $W$ of the unit ball intersecting the unit sphere, one can find an element $y\in W$ and a modulus one scalar $θ$ such that $\|x+θy\|$ is almost two. It is known that spaces with the Daugavet property satisfy this condition, and that this condition implies the Alternative Daugavet property. We first provide examples of super ADP spaces which fail the Daugavet property. We show that the norm of a super ADP space is rough, hence the space cannot be Asplund, and we also prove that the space fails the point of continuity property (particularly, the Radon--Nikodým property). In particular, we get examples of spaces with the Alternative Daugavet property that fail the super ADP. For a better understanding of the differences between the super ADP, the Daugavet property, and the Alternative Daugavet property, we will also consider the localizations of these three properties and prove that they behave rather differently. As a consequence, we provide characterizations of the super ADP for spaces of vector-valued continuous functions and of vector-valued integrable functions.

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arXiv2024

Slicely countably determined points in Banach spaces

We introduce slicely countably determined points (SCD points) of a bounded and convex subset of a Banach space which extends the notions of denting points, strongly regular points and much more. We completely characterize SCD points in the unit balls of $L_1$-preduals. We study SCD points in direct sums of Banach spaces and obtain that an infinite sum of Banach spaces may have an SCD point despite the fact that none of its components have it. We then prove sufficient conditions to get that an elementary tensor $x\otimes y$ is an SCD point in the unit ball of the projective tensor product $X \widehat{\otimes}_πY$. Regarding Lipschitz-free spaces on compact metric spaces, we show that norm one SCD points of their unit balls are exactly the ones that can be approximated by convex combinations of strongly exposed points of the unit ball. Finally, as applications, we prove a new inheritance result for the Daugavet property to its subspaces, we show that separable Banach spaces for which every convex series of slices intersects the unit sphere must contain an isomorphic copy of $\ell_1$, and we get pointwise conditions on an operator on a Banach space with the Daugavet property to satisfy the Daugavet equation.

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arXiv2023

Unconditional bases and Daugavet renormings

We introduce a new diametral notion for points of the unit sphere of Banach spaces, that naturally complements the notion of Delta-points, but is weaker than the notion of Daugavet points. We prove that this notion can be used to provide a new geometric characterization of the Daugavet property, as well as to recover -- and even to provide new -- results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Finally, we show that this notion leads to powerful new ideas for renorming questions, and that those ideas can be combined with previous constructions from the literature in order to renorm every infinite dimensional Banach space with an unconditional Schauder basis to have a Daugavet point.

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arXiv2022

Transfinite almost square Banach spaces

It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of $c_0(κ)$, where $κ$ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we obtain a solution to an open question from [8].

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arXiv2021

Stability of diametral diameter two properties

We prove that the diametral diameter two properties are inherited by $F$-ideals (e.g., $M$-ideals). On the other hand, these properties are lifted from an $M$-ideal to the superspace under strong geometric assumptions. We also show that all of the diametral diameter two properties are stable under the formation of corresponding Köthe-Bochner spaces (e.g., $L_p$-Bochner spaces). Finally, we investigate when the projective tensor product of two Banach spaces has some diametral diameter two property.

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arXiv2021

Attaining strong diameter two property for infinite cardinals

We extend the (attaining of) strong diameter two property to infinite cardinals. In particular, a Banach space has the 1-norming attaining strong diameter two property with respect to $ω$ (1-ASD2P$_ω$ for short) if every convex series of slices of the unit ball intersects the unit sphere. We characterize $C(K)$ spaces and $L_1(μ)$ spaces having the 1-ASD2P$_ω$. We establish dual implications between the 1-ASD2P$_ω$, $ω$-octahedral norms and Banach spaces failing the $(-1)$-ball-covering property. The stability of these new properties under direct sums and tensor products is also investigated.

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arXiv2021

A characterization of Banach spaces containing $\ell_1(κ)$ via ball-covering properties

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 $\ell_1(κ^+)$ if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by $κ$ many open balls not containing $αB_X$, where $α\in (0,1)$. We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.

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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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arXiv2020

Octahedral norms in duals and biduals of Lipschitz-free spaces

We continue with the study of octahedral norms in the context of spaces of Lipschitz functions and in their duals. First, we prove that the norm of $\mathcal F(M)^{**}$ is octahedral as soon as $M$ is unbounded or is not uniformly discrete. Further, we prove that a concrete sequence of uniformly discrete and bounded metric spaces $(K_m)$ satisfies that the norm of $\mathcal F(K_m)^{**}$ is octahedral for every $m$. Finally, we prove that if $X$ is an arbitrary Banach space and the norm of $\operatorname{Lip}_0(M)$ is octahedral, then the norm of $\operatorname{Lip}_0(M,X^\ast)$ is octahedral. These results solve several open problems from the literature.

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arXiv2020

Symmetric strong diameter two property in tensor products of Banach spaces

We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also prove a result for the symmetric strong diameter two property for the injective tensor product.

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arXiv2019

Bidual octahedral renormings and strong regularity in Banach spaces

We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its unit ball has diameter two).

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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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arXiv2017

Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums

We prove that, if Banach spaces $X$ and $Y$ are $δ$-average rough, then their direct sum with respect to an absolute norm $N$ is $δ/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ the space $X\oplus_p Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for any $δ$ in $(1,2]$ there is a Banach space which is exactly $δ$-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and $\infty$-sums.

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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

Rough norms in spaces of operators

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that $\mathcal L(X,Y)$ is $δ$-average rough whenever $X^\ast$ is $δ$-average rough and $Y$ is alternatively octahedral. This allows us to give a unified improvement of two theorems by Becerra Guerrero, López-Pérez, and Rueda Zoca [J. Math. Anal. Appl. 427 (2015)].

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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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