Existence and examples of quantum isometry groups for a class of compact metric spaces

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X,d)$(X,d)$ 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)$(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space (X,d)$(X,d)$ which admits a one-to-one continuous map f:X→Rⁿ$f:X\to {\mathbb{R}}^n$ for some n   such that _d0(f(x),f(y))=?(d(x,y))$d_0(f(x),f(y))=?(d(x,y))$ (where _d0$d_0$ is the Euclidean metric) for some homeomorphism ?   of R⁺${\mathbb{R}}^{+}$.