Geodesic
The Generalized Pompeiu Metric in the Isometry Problem for Hyperspaces
Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 163-170

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The isometries of the hyperspace of a compact subset of the real line endowed with the generalized Pompeiu metric are considered. It is proved that any such an isometry is generated by an isometry of the base space. 
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     author = {V. V. Aseev and A. V. Tetenov and A. P. Maksimova},
     title = {The {Generalized} {Pompeiu} {Metric} in the {Isometry} {Problem} for {Hyperspaces}},
     journal = {Matemati\v{c}eskie zametki},
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V. V. Aseev; A. V. Tetenov; A. P. Maksimova. The Generalized Pompeiu Metric in the Isometry Problem for Hyperspaces. Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 163-170. http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a0/