Related papers: On Daugavet indices of thickness

In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented…

A diversity $\delta$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0,\infty)$, with the properties that $\delta(X)=0$ if and only if $|X|\leq 1$ and $\delta(X\cup Y)\leq\delta(X\cup Z)+\delta(Z\cup Y)$,…

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…

Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad…

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some…

This paper addresses the study and characterizations of variational convexity of extended-real-valued functions on Banach spaces. This notion has been recently introduced by Rockafellar, and its importance has been already realized and…

For a bounded metric space $ X $ one can consider the quantity $ \delta(X) := \text{inf\rule[-0.5ex]{0em}{1ex}}_{\,p\in X}\; \text{sup}_{q \in X} \; d(p,q) $. This purely metric invariant is known from approximation theory as the relative…

A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id + T\|= 1 + \|T\|$ holds for every rank-one operator $T:X \longrightarrow X$. We show that the most natural attempts to introduce new properties by considering…

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property.…

This work investigates dense packings of congruent hard infinitesimally--thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $\theta \in \left ( \pi, 2\pi \right ]$.…

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed…

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a…

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for…

The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains…

The paper is devoted to the convex-set counterpart of the theory of weak$^*$ derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space $X$ and every countable…

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we…

We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly…

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces $X$ and $Y$, the Kadets distance is defined to be the infimum of the Hausdorff distance $d(B_X,B_Y)$ between the respective…

Let $M$ be a metric space and $X$ be a Banach space. In this paper we address several questions about the structure of $\mathcal F(M)\widehat{\otimes}_\pi X$ and $\mathop{Lip}(M,X)$. Our results are the following: (1) We prove that if $M$…

For Banach spaces of analytic functions on the disc for which the polynomials are dense and their pointt evaluations continuous, we prove the following: If they contain a function such that the limit superior of its modulus is infinite…