Related papers: Daugavet- and delta-points in Banach spaces with u…
For a compact metric space $K$ the space $\Lip(K)$ has the Daugavet property if and only if the norm of every $f \in \Lip(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<\infty$, this is equivalent to the convexity of…
We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Gin\'es L\'opez-P\'erez and Abraham Rueda Zoca in J. Conv. Anal. \textbf{25}, no. 3 (2018).
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…
The concept of $\ell_{\Phi}$-decomposition, extending the concept of $\ell_{p}$-decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and $\ell_{\Phi}$-decompositions are studied. We show that…
We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem,…
It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and…
A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator…
A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every \ep > 0, there exists \delta > 0 such that for every \ep-separated sequence (x_n) in the unit ball B, and every x in B, there are…
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…
An operator $G : \allowbreak X \to Y$ is said to be a Daugavet center if $\|G + T\| = \|G\| + \|T\|$ for every rank-1 operator $T : \allowbreak X \to Y$. The main result of the paper is: if $G : \allowbreak X \to Y$ is a Daugavet center,…
The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The…
We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two…
We obtain new progresses about the diameter two property and the Daugavet property in tensor product spaces. Namely, the main results of the paper are: -If $X^*$ has the WODP, then $X\widehat{\otimes}_\varepsilon Y$ has the DD2P for any…
A generalization of Lozanovskii's result is proved. Let E be $k$-dimensional subspace of an $n$-dimensional Banach space with unconditional basis. Then there exist $x_1,..,x_k \subset E$ such that $B_E \p \subset \p absconv\{x_1,..,x_k\}$…
We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or $X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for…
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…
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…
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…
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…
We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of…