Related papers: Slicely Countably Determined 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…
In this note, we study the geometry of the unit ball of the Banach space generated by the adequate family of all subsets of branches of the infinite binary tree, and answer several open questions related to slicely countably determined…
We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of…
A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy…
We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former…
We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact…
In this article, we study the Daugavet property and the diametral diameter two properties in complex Banach spaces. The characterizations for both Daugavet and $\Delta$-points are revisited in the context of complex Banach spaces. We also…
We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable…
We introduce extensions of $\Delta$-points and Daugavet points in which slices are replaced by relative weakly open subsets (super $\Delta$-points and super Daugavet points) or by convex combinations of slices (ccs $\Delta$-points and ccs…
We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index~1 are equivalent. In the class of rearrangement invariant (r.i.)\ sequence spaces the only examples of spaces with these…
We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius…
One shows for Banach bundles in a certain class that having a second countable locally compact Hausdorff base space and separable fibers implies the separability of the Banach space of the all sections that vanish at infinity. In the…
We construct a Banach space $X$ with the r-BSP such that the infimum of the diameter of the slices of the unit ball is $1$, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by…
Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected…
Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analysed. A geometric description is given in terms of nonlinear slices. Some local versions of…
We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm…
We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is:…
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…
All most all the function spaces over real or complex domains and spaces of sequences, that arise in practice as examples of normed complete linear spaces (Banach spaces), are reflexive. These Banach spaces are dual to their respective…
We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit…