The Lipschitz distance between two metric spaces is defined by
$$d_{\mathcal L}(X, Y) = \inf_f \log(\{\max\{\text{dil}(f), \text{dil}(f^{-1})\})$$
where
$$\text{dil}(f) = \sup_{x_1, x_2 \in X} \frac{ d_Y(f(x_1), f(x_2)) } {d_X(x_1, x_2)}$$
and the infimum runs over bi-Litschitz homeomorphisms between $X$ and $Y$. In my notes I have it written that this defines a metric on the isometry classes of compact metric spaces.
I am having trouble understanding why identity of the indiscernables holds, i.e.
$$d_{\mathcal L}(X, Y) = 0 \Longleftrightarrow X \text{ and } Y \text{ are isometric}$$
A case where there might not be an isometry between $X$ and $Y$ is where the infimum $0$ is not attained on the set of homeomorphisms between $X$ and $Y$. Why can't this be the case? Is the set of all homeomorphisms between $X$ and $Y$ compact also? If so, why?
Thanks in advance!
The Ascoli Arzela theorem tells you exactly what you need: from a sequence of bilipschitz homeomorphisms $f_i$ for which $dil(f)$ approaches 1, you extract a subsequence converging to one with dilation equal to 1.
It's not true that the set of homeomorphisms $X\to Y$ is compact, but the magic of Ascoli Arzela is a very useful criterion on sequences of functions---equicontinuity--under which a convergent subsequence must exist. So all you need is to check that equicontinuity holds for the sequence $f_i$, and then to verify that a sub sequential limit is what it's supposed to be.