Isometric embeddings of finite metric spaces

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• 作者：A. I. Oblakova
• 刊名：Moscow University Mathematics Bulletin
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：71
• 期：1
• 页码：1-6
• 全文大小：537 KB
• 参考文献：1.S. D. Iliadis, “A separable complete metric space of dimension n containing isometrically all compact metric space of dimension n,” topol. and its appl., 160 (11), 1271 (2013).MathSciNet CrossRef MATH
  2.S. D. Iliadis and I. Naidoo, “On Isometric Embeddings of Compact Metric Spaces of a Countable Dimension,” Topol, and its Appl. 160 (11), 1284 (2013).MathSciNet CrossRef MATH
  3.J. Tits, “Groupes a Croissance Polynomiale,” in Seminaire Bourbaki, 1980-1981 (Springer, Berlin Heidelberg, 1981), pp. 176–188.
  4.S. D. Iliadis, Universal Spaces and Mappings (Elsevier Science, Amsterdam, 2005).MATH
  5.R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989; Mir, Moscow, 1986).MATH
  6.W. Hurewicz and H. Wallman, Dimension Theory (Princeton Univ. Press, Princeton, N.J., 1941; Inostr. Liter., Moscow, 1948).MATH
  
• 作者单位：A. I. Oblakova (1)
  
  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Russian Library of Science
  
• 出版者：Allerton Press, Inc. distributed exclusively by Springer Science+Business Media LLC
• ISSN：1934-8444

文摘

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R m that isometrically contains all finite metric spaces which can be embedded into R m , contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R m and, in particular, for the Euclidean metric.