Related papers: Plasticity of the unit ball of $c$ and $c_0$

We show that if $K$ is a compact metrizable space with finitely many accumulation points, then the closed unit ball of $C(K)$ is a plastic metric space, which means that any non-expansive bijection from $B_{C(K)}$ onto itself is in fact an…

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are…

A metric space is plastic if all its non-expansive bijections are isometries. We prove three main results: (1) every countable dense subspace of a normed space is not plastic, (2) every $k$-crowded separable metric space contains a plastic…

It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E…

Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces…

We characterize the surjective isometries, with respect to the Hausdorff distance, of the class of bodies given by intersections of Euclidean unit balls. We show that any such isometry is given by the composition of a rigid motion with…

We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also…

This paper demonstrates the expand-contract plasticity of the unit spheres of $\ell_1$, $\ell_{\infty}$, and $c$. Furthermore, it establishes the strong plasticity of the unit spheres of Hilbert spaces.

We show that the unit ball of a Hilbert space in its weak topology is a continuous image of the countable power of the Alexandroff compactification of a discrete set, and we deduce some combinatorial properties of its lattice of open sets…

We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.

We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by…

We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an…

The article deals with plastic and non-plastic sub-spaces $A$ of the real line ${\mathbb{R}}$ with the usual Euclidean metric $d$. It investigates non-expansive bijections, proves properties of such maps and demonstrates their relevance by…

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a…

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if…

We show that every nonvoid relatively weakly open subset, in particular every slice, of the unit ball of an infinite-dimensional uniform algebra has diameter~2.

We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $\epsilon_0$ depending only…

We study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in Banach spaces. We show that every Banach space containing isomorphic copies of $c_0$ can be equivalently renormed so that…

Given a closed subset $\La$ of the open unit ball $B_1\subset \real^n$, $n \geq 3$, we will consider a complete Riemannian metric $g$ on $\bar{B_1} \setminus \La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the…

In Euclidean spaces, every closed, bounded, convex set can be characterized by two equivalent notions of separation properties. This is not true in general for arbitrary Banach spaces. In this work, we present a ball separation…