Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 1019-1064
Institute of Applied Mathematics and Mechaniks of NASU,
Function Theory Department
Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine;
victoriiabilet 'at' gmail.com
Institute of Applied Mathematics and Mechaniks of NASU,
Function Theory Department
Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine;
oleksiy.dovgoshey 'at' gmail.com
Mersin University, Faculty of Art and Sciences,
Department of Mathematics
Mersin 33342, Turkey; mkucukaslan 'at' mersin.edu.tr
Institute of Applied Mathematics and Mechaniks of NASU,
Function Theory Department
Dobrovolskogo str. 1, Slovyansk, 84100, Ukraine;
eugeniy.petrov 'at' gmail.com
Abstract. Let M be a class of metric spaces. A metric space Y is minimal M-universal if every X ∈ M can be isometrically embedded in Y but there are no proper subsets of Y satisfying this property. We find conditions under which, for given metric space X, there is a class M of metric spaces such that X is minimal M-universal. We generalize the notion of minimal M-universal metric space to notion of minimal M-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class M is minimal M-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and n-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces X which possesses the following property. Among every three distinct points of X there is one point lying between the other two points.
2010 Mathematics Subject Classification: Primary 54E35, 30L05, 54E40, 51F99.
Key words: Metric space, isometric embedding, universal metric space, betweenness relation in metric spaces.
Reference to this article: V. Bilet, O. Dovgoshey, M. Küçükaslan and E. Petrov: Minimal universal metric spaces. Ann. Acad. Sci. Fenn. Math. 42 (2017), 1019-1064.
https://doi.org/10.5186/aasfm.2017.4261
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