On unique determination of domains in Euclidean spaces
 A. P. Kopylov
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The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction.
The foundations of the second direction are presented in the second part of the paper, i.e., in Sec. 8, for the first time. This direction is closely related with the first one and consists of studying the problem of unique determination of conformal type. The main result of the section is the theorem on the unique determination of bounded convex domains by relative conformal moduli of their boundary conductors.
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 Title
 On unique determination of domains in Euclidean spaces
 Journal

Journal of Mathematical Sciences
Volume 153, Issue 6 , pp 869898
 Cover Date
 20080901
 DOI
 10.1007/s1095800891495
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Springer US
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 Authors

 A. P. Kopylov ^{(1)}
 Author Affiliations

 1. Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Moscow, Russia