相关论文: Daugavet and diameter two properties in Orlicz-Lor…
We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets…
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 two new notions called the Daugavet constant and $\Delta$-constant of a point, which measure quantitatively how far the point is from being Daugavet point and $\Delta$-point and allow us to study Daugavet and $\Delta$-points in…
We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of…
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of…
In this paper, we study the noncommutative Orlicz space $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$, which generalizes the concept of noncommutative $L^{p}$ space, where $\mathcal{M}$ is a von Neumann algebra, and $\varphi$ is an Orlicz…
Several local geometric properties of Orlicz space $L_\phi$ are presented for an increasing Orlicz function $\phi$ which is not necessarily convex, and thus $L_\phi$ does not need to be a Banach space. In addition to monotonicity of $\phi$…
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…
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 characterise the weak$^*$ symmetric strong diameter $2$ property in Lipschitz function spaces by a property of its predual, the Lipschitz-free space. We call this new property decomposable octahedrality and study its duality with the…
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 $\omega$ (1-ASD2P$_\omega$ for short) if every convex…
We prove that in Lipschitz-free spaces the strong diameter two property, the diameter two property, and the local diameter two property coincide with their corresponding attaining variants.
In this note we study the inheritance of the slice diameter two property by ultrapower spaces. Given a Banach space $X$, we give a characterisation of when $(X)_\mathcal U$, the ultrapower of $X$ through a free ultrafilter $\mathcal U$, has…
We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $ K $ of a~finite-dimensional Banach space $ (\mathbf{R}^{n}, \phi) $, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $ \phi $.…
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $\varphi:\…
We study the Daugavet property in tensor products of Banach spaces. We show that $L_1(\mu)\widehat{\otimes}_\varepsilon L_1(\nu)$ has the Daugavet property when $\mu$ and $\nu$ are purely non-atomic measures. Also, we show that…
In this paper the necessary and sufficient conditions were given for Orlicz-Lorentz function space endowed with the Orlicz norm having non-squareness and local uniform non-squareness.
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…
Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikod\'ym property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to…
We show that $L^p$-spaces over atomless measure spaces can be characterized in terms of a $p$-concavity type geometric property that is related with the Daugavet property.