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Metric preserving functions and Gromov-Hausdorff space

Last modified: 2023-05-15

#### Abstract

This work is devoted to the study of deformations of metric spaces induced by metric preserving functions. In this work, we show that continuous metric preserving functions correctly define the maps of the Gromov-Hausdorff space to themselves, and these maps have a number of interesting properties, in particular they are continuous, and they are Lipschitz if and only if the corresponding metric preserving functions are Lipschitz. We also study the deformations of arbitrary metrics defined by metric preserving functions depending on the parameter, and we provide a criterion for the continuity of the lengths of the curves.