Definition:Hausdorff Metric

Definition

Let $M = \struct {S, d}$ be a metric space.

Let $\CC$ be the set of compact subsets of $M$.

Let $\d: \CC \times \CC \to \R_{\ge 0}$ be the distance function on $M$ defined as:

$\forall A, B \in \CC: \map \d {A, B} :=$ the smallest $r \in \R_{\ge 0}$ such that $A$ and $B$ are each contained within the $r$-neighborhood of the other.

Then $\d$ is known as the Hausdorff metric on $M$.

Also known as

The Hausdorff metric is also known as the Hausdorff distance.

Also see

• Metric Space Induced by Hausdorff Metric

• Results about the Hausdorff metric can be found here.

Source of Name

This entry was named for Felix Hausdorff.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hausdorff metric