Geodesic
On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics.~II
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 986-993

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We prove the theorem on the unique determination of a strictly convex domain in $\mathbb R^n$, where $n \ge 2$, in the class of all $n$-dimensional domains by the condition of the local isometry of the Hausdorff boundaries in the relative metrics, which is a generalization of A. D. Aleksandrov's theorem on the unique determination of a strictly convex domain by the condition of the (global) isometry of the boundaries in the relative metrics. We also prove that, in the cases of a plane domain $U$ with nonsmooth boundary and of a three-dimensional domain $A$ with smooth boundary, the convexity of the domain is no longer necessary for its unique determination by the condition of the local isometry of the boundaries in the relative metrics.
Keywords: intrinsic metric, relative metric of the boundary, local isometry of the boundaries, strict convexity. 
@article{SEMR_2017_14_a63,
     author = {A. P. Kopylov},
     title = {On the unique determination of domains by the condition of the local isometry of the boundaries in the relative {metrics.~II}},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {986--993},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a63/}
}
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A. P. Kopylov. On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics.~II. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 986-993. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a63/