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Hausdorff Mapping: 1-Lipschitz and Isometry Properties

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Abstract

Properties of the Hausdorff mapping H taking each compact metric space to the space of its non-empty closed subsets endowed with the Hausdorff metric are studied. It is shown that this mapping is non-stretching (i.e., a Lipschitz mapping whose Lipschitz constant equals 1). Several examples of classes of metric spaces such that the distances between them are preserved by the mapping H are given. The distance between any connected metric space with a finite diameter and any simplex with a greater diameter is calculated. Some properties of the Hausdorff mapping are discussed, which may be useful to understand whether the mapping H is isometric or not.

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References

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Authors and Affiliations

  1. Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119991, Russia

    I. A. Mikhailov

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  1. I. A. Mikhailov
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Correspondence to I. A. Mikhailov.

Additional information

Original Russian Text © I.A. Mikhailov, 2018, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2018, Vol. 73, No. 6, pp. 3–8.

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Mikhailov, I.A. Hausdorff Mapping: 1-Lipschitz and Isometry Properties. Moscow Univ. Math. Bull. 73, 211–216 (2018). https://doi.org/10.3103/S0027132218060013

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  • DOI: https://doi.org/10.3103/S0027132218060013

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