We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banicas definition for finite metric spaces. For metric spaces $(X,d)$ which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space $(X,d)$ which admits a one-to-one continuous map $f : X raro IR^n$ for some $n$ such that $d_0(f(x),f(y))=phi(d(x,y))$ (where $d_0$ is the Euclidean metric) for some homeomorphism $phi$ of $IR^+$. As concrete examples, we obtain Wangs quantum permutation group $cls_n^+$ and also the free wreath product of $IZ_2$ by $cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in cite{huang1}.
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