Geodesic
A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains
Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 52-96

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We say that a domain $U\subset\mathbb R^n$ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain $V\subset\mathbb R^n$ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of $U$, is isometric to $U$ in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary. 
@article{MT_2009_12_2_a2,
     author = {M. V. Korobkov},
     title = {A criterion for the unique determination of domains in {Euclidean} spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains},
     journal = {Matemati\v{c}eskie trudy},
     pages = {52--96},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2009_12_2_a2/}
}
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M. V. Korobkov. A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains. Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 52-96. http://geodesic.mathdoc.fr/item/MT_2009_12_2_a2/