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Plastic pairs of metric spaces
... This property and its uniform version were researched in the study mentioned in the reference [5]. It says that any expansion of a distance between two points implies the existence of two other points which are contracted by the map ϕ. ...
The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d . It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.
This front matter of the book is free downloadable from the publisher web-cite https://link.springer.com/book/10.1007%2F978-3-319-92004-7#toc
It is known that if M is a finite-dimensional Banach space, or a strictly convex space, or the space , then every non-expansive bijection is an isometry. We extend these results to non-expansive bijections between unit balls of two different Banach spaces. Namely, if E is an arbitrary Banach space and M is finite-dimensional or strictly convex, or the space then every non-expansive bijection is an isometry.
https://rdcu.be/6Cru
Let X be a strictly convex Banach space, and let be its unit ball. Then every non-expansive bijection is an isometry.
We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.
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.
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- W Nitka
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