Symmetric products of the Euclidean spaces and the spheres

Naotsugu Chinen

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 2, page 209-221
  • ISSN: 0010-2628

Abstract

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By F n ( X ) , n ≥ 1 , we denote the n -th symmetric product of a metric space ( X , d ) as the space of the non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric d H . In this paper we shall describe that every isometry from the n -th symmetric product F n ( X ) into itself is induced by some isometry from X into itself, where X is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the n -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and present that the 2 nd symmetric product of the plane is bi-Lipschitz equivalent to the 4-dimensional Euclidean space.

How to cite

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Chinen, Naotsugu. "Symmetric products of the Euclidean spaces and the spheres." Commentationes Mathematicae Universitatis Carolinae 56.2 (2015): 209-221. <http://eudml.org/doc/270140>.

@article{Chinen2015,
abstract = {By $F_n(X)$, $n \ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n(X)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and present that the $2$nd symmetric product of the plane is bi-Lipschitz equivalent to the 4-dimensional Euclidean space.},
author = {Chinen, Naotsugu},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere; isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere},
language = {eng},
number = {2},
pages = {209-221},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Symmetric products of the Euclidean spaces and the spheres},
url = {http://eudml.org/doc/270140},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Chinen, Naotsugu
TI - Symmetric products of the Euclidean spaces and the spheres
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 2
SP - 209
EP - 221
AB - By $F_n(X)$, $n \ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n(X)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and present that the $2$nd symmetric product of the plane is bi-Lipschitz equivalent to the 4-dimensional Euclidean space.
LA - eng
KW - isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere; isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere
UR - http://eudml.org/doc/270140
ER -

References

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