Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, Issue 3(22), Pages 34–40 (Mi vvgum52)  

Mathematics

On history of relative distance in plane domain

Yu. V. Pomelnikov

Volgograd State University
References:
Abstract: In this paper we consider history of relative distance in plane domain and its applications. Definitions of relative distance in works S. Mazurkevich, M. Lavrentiev, G. Suvorov and V. Miklukov are explained and compared with its role in geometric theory of functions.
The definitions of relative distance are discussed in this paper. Relative distance is defined by Marzinkevich (1916) as diameter of set connected at two points. This distance cannot be considered as prime ends distance. Lavrentiev's distance (1936) is defined as minimum of two figures length of curves connected at two points and length of cuts separating two points from fixed point. This variant of distance completes domain with prime ends but does not meets triangle inequality. Definitions given by Suvorov (1956) and recently by Miklyukov (2004) meet all conditions. Example of plain domain where triangle inequality for Lavrentiev's distance fails are presented in this paper.
Keywords: relative distance, prime ends, boundary correspondence, compactification, triangle inequality.
Document Type: Article
UDC: 517.544.7
BBC: 22.1
Language: Russian
Citation: Yu. V. Pomelnikov, “On history of relative distance in plane domain”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, no. 3(22), 34–40
Citation in format AMSBIB
\Bibitem{Pom14}
\by Yu.~V.~Pomelnikov
\paper On history of relative distance in plane domain
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2014
\issue 3(22)
\pages 34--40
\mathnet{http://mi.mathnet.ru/vvgum52}
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