Hausdorff distance

Definition

Suppose $A$ and $B$ are two subsets of a metric space $(X,d)$. The Hausdorff distance between $A$ and $B$, denoted $d_H(A,B)$, is defined as:

$d_H(A,B)=inf\{\epsilon \mid A\in B_{\epsilon }(B),B\in B_{\epsilon }(A)\}$.

Here $B_{\epsilon }(M)$ denotes the union of balls of radius $\epsilon$ about all points in $M$.

Related notions

• Gromov-Hausdorff distance

Facts

• Hausdorff distance is a metric on closed subsets of a compact metric space