Abstract
The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.
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Gruber, P.M., Tichy, R. Isometries of spaces of compact or compact convex subsets of metric manifolds. Monatshefte für Mathematik 93, 117–126 (1982). https://doi.org/10.1007/BF01301399
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DOI: https://doi.org/10.1007/BF01301399