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:

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

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

Related notions

 * Gromov-Hausdorff distance

Facts

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