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A Universal Separable Diversity

100%
Bryant D., Nies A., Tupper P.
Analysis and Geometry in Metric Spaces
|
2017
|
tom 5
|
nr 1
138-151
EN
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.
2
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The Demyanov metric and some other metrics in the family of convex sets

100%
Rzeżuchowski T.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2229-2239
EN
We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.
3
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Fixed points for cyclic orbital generalized contractions on complete metric spaces

84%
Karapınar E., Romaguera S., Taş K.
Open Mathematics
|
2013
|
tom 11
|
nr 3
552-560
EN
We prove a fixed point theorem for cyclic orbital generalized contractions on complete metric spaces from which we deduce, among other results, generalized cyclic versions of the celebrated Boyd and Wong fixed point theorem, and Matkowski fixed point theorem. This is done by adapting to the cyclic framework a condition of Meir-Keeler type discussed in [Jachymski J., Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl., 1995, 194(1), 293–303]. Our results generalize some theorems of Kirk, Srinavasan and Veeramani, and of Karpagam and Agrawal.
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