Plasticity in metric spaces

Abstract

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.  2005 Elsevier Inc. All rights reserved.

Nonexpansive Bijections to the Unit Ball of the -Sum of Strictly Convex Banach Spaces

“…Observe, that U n and V n are isometric to the unit ball of the n-dimensional 1 , so they can be considered as two copies of the same compact metric space. The expand-contract plasticity of totally bounded metric spaces [5] implies that every bijective nonexpansive map from U n onto V n is an isometry. In particular, F maps U n onto V n isometrically.…”

Plasticity of the unit ball of $c$ and $c_0$

“…We fix a function σ : N → N and a sequence α : N → {−1, 1}. We show that for each n ∈ N there exists a continuous function 1]. We also know that if α n = 1, then f n is strictly increasing, f n (−1) < 0, f n (0) = 0 and f n (1) > 0, and if α n = −1, then f n is strictly decreasing, f n (−1) > 0, f n (0) = 0 and f n (1) < 0.…”

“…We call a metric space plastic if every non-expansive bijection from the space onto itself is an isometry. The concept was introduced in [1] by S. A. Naimpally, Z. Piotrowski and E. J. Wingler. It seems that the class of plastic metric spaces does not have a simple characterization.…”

Two new examples of Banach spaces with a plastic unit ball

“…We call a metric space X plastic if every non-expansive bijection from X onto itself is an isometry. The last notion was introduced by S. A. Naimpally, Z. Piotrowski, and E. J. Wingler in [8]. It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1].…”

“…It is known that every totally bounded metric space is plastic, see [3,Satz IV] or [8,Theorem 1.1]. On the other hand, a plastic metric space need not be totally bounded nor bounded -e.g., the set of integers with the usual metric is plastic [8,Theorem 3.1]. There are also examples of bounded metric spaces that are not plastic, one of our favorite examples here is a solid ellipsoid in Hilbert space ℓ 2 (Z) with infinitely many semi-axes equal to 1 and infinitely many semi-axes equal to 2, see [1,Example 2.7].…”