Hausdorff Mapping: 1-Lipschitz and Isometry Properties / I. A. Mikhailov. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2018. № 6. P. 3-8 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 211-216].

Properties of the Hausdorff mapping \mathcal{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 \mathcal{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 \mathcal{H} is isometric or not.

Key words: Hausdorff distance, Gromov–Hausdorff space, Hausdorff mapping.

№ 6/2018