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 Mat. Tr., 2009, Volume 12, Number 2, Pages 52–96 (Mi mt181)

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

M. V. Korobkovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: 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.

Key words: domain, Hausdorff limit, relative metric, intrinsic metric, uniqueness of determination.

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English version:
Siberian Advances in Mathematics, 2010, 20:4, 256–284

Bibliographic databases:

UDC: 514.772.35

Citation: 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”, Mat. Tr., 12:2 (2009), 52–96; Siberian Adv. Math., 20:4 (2010), 256–284

Citation in format AMSBIB
\Bibitem{Kor09} \by M.~V.~Korobkov \paper 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 \jour Mat. Tr. \yr 2009 \vol 12 \issue 2 \pages 52--96 \mathnet{http://mi.mathnet.ru/mt181} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2599425} \elib{http://elibrary.ru/item.asp?id=13021587} \transl \jour Siberian Adv. Math. \yr 2010 \vol 20 \issue 4 \pages 256--284 \crossref{https://doi.org/10.3103/S1055134410040024} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. Y. Novak, “Lower bounds to the accuracy of sample maximum estimation”, Theory Stoch. Process., 15(31):2 (2009), 156–161
2. Anatoly P. Kopylov, Mikhail V. Korobkov, “Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold”, Zhurn. SFU. Ser. Matem. i fiz., 9:3 (2016), 320–331
3. A. P. Kopylov, “On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics”, Sib. elektron. matem. izv., 14 (2017), 59–72
4. A. P. Kopylov, “On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. II”, Sib. elektron. matem. izv., 14 (2017), 986–993
5. Kopylov A.P., “Problems of Unique Determination of Domains By the Relative Metrics on Their Boundaries”, Lobachevskii J. Math., 38:3, SI (2017), 476–487
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