Geodesic
Symmetric products of the Euclidean spaces and the spheres
Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 2, pp. 209-221.

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By $F_n(X)$, $n \geq 1$, 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.
DOI : 10.14712/1213-7243.2015.118
Classification : 30C65, 30L10, 54B10, 54B20, 54E35
Keywords: isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere 
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Chinen, Naotsugu. Symmetric products of the Euclidean spaces and the spheres. Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 2, pp. 209-221. doi : 10.14712/1213-7243.2015.118. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.118/

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