Symmetric products of the Euclidean spaces and the spheres

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