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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

Authors
  • Korobkov, M. V.1
  • 1 Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia , Novosibirsk (Russia)
Type
Published Article
Journal
Siberian Advances in Mathematics
Publisher
Allerton Press, Inc.
Publication Date
Oct 01, 2010
Volume
20
Issue
4
Pages
256–284
Identifiers
DOI: 10.3103/S1055134410040024
Source
Springer Nature
Keywords
License
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Abstract

We say that a domain U ⊂ ℝn 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 ⊂ ℝn 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.

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