Geodesic
On Hausdorff intrinsic metric
Lobachevskii journal of mathematics, Tome 8 (2001), pp. 185-189

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In this paper we prove that in the set of all nonempty bounded closed subsets of a metric space $(X,\rho)$ the Hausdorff metric is the Hausdorff intrinsic metric if and only if the metric $\rho$ is an intrinsic metric. In a space with an intrinsic metric we obtain the upper bound for the Hausdorff distance between generalized balls. 
@article{LJM_2001_8_a4,
     author = {E. N. Sosov},
     title = {On {Hausdorff} intrinsic metric},
     journal = {Lobachevskii journal of mathematics},
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     volume = {8},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2001_8_a4/}
}
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E. N. Sosov. On Hausdorff intrinsic metric. Lobachevskii journal of mathematics, Tome 8 (2001), pp. 185-189. http://geodesic.mathdoc.fr/item/LJM_2001_8_a4/