Hausdorff distance

Remarks

Equivalently, $d_H(A, B)$ is the infimum of the $\varepsilon \ge 0$ with $A \subseteq B_\varepsilon$ and $B \subseteq A_\varepsilon$, where $S_\varepsilon = \set{x \in M : d(x, S) \le \varepsilon}$. In $\R^n$ with the Euclidean distance and closed unit ball $D$, this reads $d_H(A, B) = \inf\set{\varepsilon \ge 0 : A \subseteq B + \varepsilon D, \ B \subseteq A + \varepsilon D}$ for nonempty compact $A, B$, with Minkowski sums.

Example: $A = [0, 1]$ and $B = [0, 2]$ in $\R$. Every point of $A$ lies in $B$, so $\sup_{a \in A} d(a, B) = 0$; the points of $B$ have $d(b, A) = \max(b - 1, 0)$, with supremum $1$ at $b = 2$. So $d_H(A, B) = 1$.

Non-example of a metric: on all nonempty subsets, $d_H$ is not a metric. For $A = (0, 1)$ and $B = [0, 1]$ in $\R$ one has $d_H(A, B) = 0$ although $A \neq B$, and $d_H(\set{0}, [0, \infty)) = +\infty$. On the nonempty closed bounded subsets of $M$, and in particular on the nonempty compact subsets, $d_H$ is a metric. If $M$ is complete, then so is this space of closed bounded subsets; if $M$ is compact, then so is the space of nonempty compact subsets.