Let $(X, d)$ be a metric space and define $K(X, d) = \{A \subseteq X : A \neq \emptyset$ and $A$ is compact in $(X, d)\}$. Let $d_h : K(X, d) \times K(X, d) \rightarrow \mathbb{R}$ be Hausdorff distance. We know that then $(K(X, d), d_h)$ is a metric space.
My question is this:
Find an example of a metric space $(X, d)$ so that we can determine that $(K(X, d), d_h)$ is isomorphic to some well known metric space. In particular, I want to find a metric space $(X, d)$ such that $K(X, d)$ is isomorphic to $([0, 1], d_2)$ where $d_2$ is euclidean metric on $\mathbb{[0, 1]}$. Since it seems to me that such metric space doesn't even exist (although I'm not sure how to prove it). Any example where we know the structure of $K(X, d)$ would be great.
Only example where I was able to determine the structure of $K(X,d)$ is this:
Let $X$ be a finite set, in particular $X = \{x_1,...,x_n\}$ for some $n \in \mathbb{N}$. Let $d : X \times X \rightarrow \mathbb{R}$ be a discrete metric on $X$. Then every subset of $X$ is finite so it is compact, therefore $K(X, d) = \mathcal{P}(X)$. Suppose $A,B \subseteq X$ and $A \neq B$. Without loss of generality we can suppose there exists $x \in A \setminus B$. Now for every $y \in B$ we have $d(x, y) = 1$. From this and the fact that $d$ is discrete metric we can conclude $d_h(A, B) = 1$. Since $d_h$ is a metric, if $A = B$ we have $d_h(A, B) = 0$, therefore we have shown $d_h$ is a discrete metric. It is easy to show now that $(K(X, d), d_h)$ is isomorphic to $(Y, d^{'})$ where $Y = \{y_1,...,y_{2^n}\}$ and $d'$ is discrete metric.
