Related papers: Lipschitz slices versus linear slices in Banach sp…

We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space…

We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex…

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radius of the balls is small enough. Being focused on the study of new and mild sufficient conditions for a…

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to…

This book discusses the interactions between the (nonlinear) metric structure of Banach spaces and their linear asymptotic behavior. The overarching problem is to understand how the various linear structures of a Banach space are preserved…

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincar\'e inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends…

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the…

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…

This paper deals with the interplay of the geometry of the norm and the weak topology in Banach spaces. Both dual and intrinsic connections between weak forms of rotundity and smoothness ared discussed. Weakly exposed points, weakly locally…

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…

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional…

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and…

We present and study a family of metrics on the space of compact subsets of $R^N$ (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many…

We consider the problem of isometric embedding of metric spaces to the Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly…

The aim of this note is to consider different notions for the Banach-Saks property in locally solid vector lattices as an extension for the known concepts of the Banach-Saks property in Banach lattices. We investigate relations between…

We show that the class of subspaces of c_0 is stable under Lipschitz isomorphisms. The main corollary is that any Banach space which is Lipschitz isomorphic to c_0 is linearly isomorphic to c_0.

We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of…

The standard theory of Banach spaces is built upon the notions of vector space, triangle inequality and Cauchy completeness. Here we propose a `hyperbolic' variant of this `elliptic' framework where general linear combinations are replaced…

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real…