Geodesic
On Unique Determination of Domains in Euclidean Spaces
Contemporary Mathematics. Fundamental Directions, Geometry, Tome 22 (2007), pp. 139-167.

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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 was 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 in 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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A. P. Kopylov. On Unique Determination of Domains in Euclidean Spaces. Contemporary Mathematics. Fundamental Directions, Geometry, Tome 22 (2007), pp. 139-167. http://geodesic.mathdoc.fr/item/CMFD_2007_22_a5/

[1] Aleksandrov A. D., Vnutrennyaya geometriya vypuklykh poverkhnostei, Gostekhizdat, M., L., 1948

[2] Aleksandrov V. A., “Izometrichnost oblastei v $\mathbb R^n$ i otnositelnaya izometrichnost ikh granits”, Sib. mat. zh., 25:3 (1984), 3–13 | Zbl

[3] Aleksandrov V. A., “Izometrichnost oblastei v $\mathbb R^n$ i otnositelnaya izometrichnost ikh granits, II”, Sib. mat. zh., 26:6 (1985), 3–8 | MR | Zbl

[4] Aleksandrov V. A., “Ob oblastyakh, odnoznachno opredelyaemykh otnositelnoi metrikoi svoei granitsy”, Issledovaniya po geometrii i matematicheskomu analizu, Tr. In-ta matematiki AN SSSR. Sib. otd., 7, 1987, 5–19 | MR | Zbl

[5] Aleksandrov V. A., “Odnoznachnaya opredelennost oblastei s nezhordanovymi granitsami”, Sib. mat. zh., 30:1 (1989), 3–12 | MR | Zbl

[6] Aleksandrov V. A., “Otsenka deformatsii strogo vypukloi oblasti v zavisimosti ot izmeneniya otnositelnoi metriki ee granitsy”, Sib. mat. zh., 31:5 (1990), 3–9 | MR

[7] Aleksandrov V. A., “Ob izometrichnosti mnogogrannykh oblastei, granitsy kotorykh lokalno izometrichny v otnositelnykh metrikakh”, Sib. mat. zh., 33:2 (1992), 3–9 | MR

[8] Borovikova M. K., “Ob izometrichnosti mnogougolnykh oblastei, granitsy kotorykh lokalno izometrichny v otnositelnykh metrikakh”, Sib. mat. zh., 33:4 (1992), 30–41 | MR

[9] Volkov Yu. A., “Otsenka deformatsii vypukloi poverkhnosti v zavisimosti ot izmeneniya ee vnutrennei metriki”, Ukr. geom. sb., 5/6, Izd-vo KhGU, Kharkov, 1968, 44–69 | MR

[10] Kopylov A. P., “O granichnykh znacheniyakh otobrazhenii, blizkikh k izometricheskim”, Sib. mat. zh., 25:3 (1984), 120–131 | MR | Zbl

[11] Kuzminykh A. V., “Ob izometrichnosti oblastei, granitsy kotorykh izometrichny v otnositelnykh metrikakh”, Sib. mat. zh., 26:3 (1985), 91–99 | MR

[12] Pogorelov A. V., Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, M., 1969 | MR

[13] Pogorelov A. V., Odnoznachnaya opredelennost obschikh vypuklykh poverkhnostei, Izd. AN USSR, 1951

[14] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR

[15] Reshetnyak Yu. G., Teoremy ustoichivosti v geometrii i analize, Izd-vo In-ta matematiki im. S. L. Soboleva SO RAN, Novosibirsk, 1996 | MR

[16] Saks S., Teoriya integrala, IL., M., 1949

[17] Senkin E. P., “Neizgibaemost vypuklykh giperpoverkhnostei”, Ukr. geom. sb., 12, Izd-vo KhGU, Kharkov, 1972, 131–152 | MR

[18] Trotsenko D. A., “Odnoznachnaya opredelennost ogranichennykh oblastei metrikoi granitsy, indutsirovannoi metrikoi oblasti”, Vsesoyuz. konf. po geometrii “v tselom”, Tez. dokl. (Novosibirsk, 1987), IM SO AN SSSR, Novosibirsk, 1987, 122

[19] Uitni Kh., Geometricheskaya teoriya integrirovaniya, IL, M., 1960

[20] Shvarts L., Analiz, T. 2, Mir, M., 1972

[21] Gehring F. W., Väisälä J., “The coefficients of quasiconformality of domains in space”, Acta Math., 114:1–2 (1965), 1–70 | DOI | MR | Zbl

[22] Pólya G., Szegő G., Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stud., 27, Princeton Univ. Press, Princeton, 1951 | MR | Zbl

[23] Väisälä J., Lectures on $n$-Dimensional Quasiconformal Mappings, Springer-Verlag, Berlin, Heidelberg, New York, 1971 | MR | Zbl

[24] Väisälä J., “On quasiconformal mappings in space”, Ann. Acad. Sci. Fenn. A I, 298 (1961), 1–36 | MR