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Rainis Haller

Publications and source records attributed to Rainis Haller.

At least 19 recordsLinked to original sources
arXiv2026

Weak stability and binary convex combinations of slices

We study weak openness and related geometric properties of convex combinations of slices of the unit balls of Banach spaces. We prove that, for every $m\geq 2$, properties $\mathrm{P1}^{(m)}$ and $\mathrm{CWO}^{(m)}$ are equivalent, and likewise for their closure variants. Consequently, both $\mathrm{P1}$ and $\overline{\mathrm{P1}}$ are determined by convex combinations of two slices; in particular P1 and CWO are equivalent. We construct infinite-dimensional real Banach spaces with $\mathrm{P2}^{(2)}$ but without P2, and with $\mathrm{P3}^{(2)}$ but without P3. In the latter space, every convex combination of two slices contains two points at distance 2, although the space fails the strong diameter two property. Finally, we give an equivalent norm on the real space $c_0$ for which $\overline{\mathrm{P1}}$ holds, but P1 fails. These answers several questions raised in the literature.

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arXiv2026

Plasticity of the unit ball of the real Banach space $\ell_\infty$

We prove that the closed unit ball of the real Banach space $\ell_\infty$ is plastic, that is, every non-expansive bijection from the unit ball onto itself is an isometry. The main step is to show that every non-expansive bijection of this ball maps extreme points to extreme points. This is done by using elementary coverings of the unit ball by balls of radius one. The conclusion then follows from a theorem by Fakhoury. The same argument also shows that, for arbitrary $\Gamma$, every non-expansive bijection of $B_{\ell_\infty(\Gamma)}$ preserves extreme points.

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arXiv2026

A counterexample for the Daugavet index of thickness in $\ell_1$-sums

We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies \[ T(X\oplus_1 Y)=\min\{T(X),T(Y)\}. \] We show that this equality does hold whenever one of the two summands has the Daugavet property. On the other hand, if $D$ is a Banach space with the Daugavet property and $N$ is a suitable absolute norm, then for $X=D\oplus_N D$, one has $T(X\oplus_1 X)<T(X)$.

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arXiv2025

Norm-one points in convex combinations of relatively weakly open subsets of the unit ball in the spaces $L_1(\mu,X)$

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(\mu)$, where $\mu$ is a non-zero $\sigma$-finite (countably additive non-negative) measure, norm-one elements in finite convex combinations of relatively weakly open subsets of the unit ball are interior points of these convex combinations in the relative weak topology. In this paper that result is generalised by proving that the same is true in the (real or complex) Lebesgue--Bochner spaces $L_1(\mu,X)$ where $X$ is a weakly uniformly rotund Banach space.

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arXiv2024

Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions

We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter two properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter two property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter two property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones.

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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

Diameter two properties for spaces of Lipschitz functions

We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$ has the symmetric strong diameter two property for every $n\in \mathbb{N}$, including the case of $n=2$; every local norm-one Lipschitz function is a Daugavet point.

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arXiv2021

Two new examples of Banach spaces with a plastic unit ball

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).

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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

Daugavet- and Delta-points in absolute sums of Banach spaces

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 $Δ$-points in absolute sums of Banach spaces was started.

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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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arXiv2017

On convex combinations of slices of the unit ball in Banach spaces

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).

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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

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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