Induced Hausdorff metrics on quotient spaces

Let $G$ be a group, $(M,d)$ be a metric space, $X$ be a compact subspace of $M$ and $\phi :G\times M\to M$ be a left action by homeomorphisms of $G$ on $M$. Denote $gp=f(g,p)$. The isotropy subgroup of $G$ with respect to $X$ is defined by $H_X=\{g\in G;gX=X\}$. In this work we define the induced Hausdorff metric on $G\mathrm{/}{H}_X$ by $d_X(g{H}_X,h{H}_X)=d_H(gX,hX)$, where $d_H$ is the Hausdorff distance on $M$. Let ${\widehat{d}}_X$ be the intrinsic metric induced by $d_X$. In this work we study the geometry of $(G\mathrm{/}{H}_X,d_X)$ and $(G/H_X,\hat d_X)$\mathrm{.}Inparticular,weprovethatif$G$isaLiegroup,$M$ is a differentiable manifold endowed with a metric $d$ which is locally Lipschitz equivalent to a Finsler metric, $X$ is a compact subset of $M$ and $\varphi:G \times M \rightarrow M$isasmoothleftactionbyisometriesof$G$on$M$, then $(G\mathrm{/}{H}_X,{\widehat{d}}_X)$ is $C^0$-Finsler.