In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that

$$ d(A, B) \le W(\mu|_A, \mu|_B), $$ where

- $d(A, B):= \inf_{a \in A,\;b \in B}d(x,y)$ is the distance between $A$ and $B$.
- $\mu|_A$ defined by $\mu|_A(C):= \mu(A\cap C)/\mu(A)$ is the conditional measure induced by $A$
- $W(\mu|_A,\mu|_B)$ is the Wasserstein distance between $\mu|_A$ and $\mu|_B$.

Now, consider the Hausdorff distance between $A$ and $B$, defined by $$ d_H(A,B) := \max\left(\sup_{b \in B}\inf_{a \in A}d(a,b),\sup_{a \in A}\inf_{b \in B}d(a,b)\right) = \inf\{\epsilon \ge 0 | A \subseteq B^\epsilon \text{ and } B\subseteq A^\epsilon\}, $$ where $A^\epsilon := \{x \in X | \exists a \in A\text{ with }d(a,x)\le \epsilon\} = \cup_{a \in A}\operatorname{Ball}_\epsilon(a)$ is the $\epsilon$-blowup of $A$.

# Question

Is $d_H(A,B)$ a lower bound (or upper bound) for some probability metric ? Which one ? Between what and what ?

# Edit

A user mentioned "$L^\infty$ Wasserstein distance" in the comments. This comment seems to have gone unnoticed until now. Indeed, **Exercise 36** of OTAM asks to prove that $d_H(supp(\mu),supp(\nu)) \le W_\infty(\mu,\nu)$ for every pair of measures $\mu$ and $\nu$ on a Polish space. This solves one half of my question. The other half has been solved (in the negative) by another user.