Article
MSC:
30C65,
30L10,
54B10,
54B20,
54E35 | MR 3338733 Reviewed Chinen, Naot | Zbl 06433818 | DOI: 10.14712/1213-7243.2015.118
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Keywords:
isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere
isometry; symmetric product; bi-Lipschitz maps; Euclidean space; sphere
Summary:
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.
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