Hausdorff metric

The colored sets on the left have a relatively small Hausdorff gap to the corresponding sets on the right.

The Hausdorff metric , named after the mathematician Felix Hausdorff , measures the distance between non-empty compact subsets , a metric space . ${\displaystyle \delta (A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle E}$

Clearly, two compact subsets have a small Hausdorff distance if for each element of one set there is an element of the other set to which it has a small distance.

definition

As an aid, one defines the distance between a point and a non-empty compact subset with recourse to the metric of the space as ${\displaystyle D}$ ${\displaystyle x}$ ${\displaystyle K\subseteq E}$ ${\displaystyle d}$ ${\displaystyle E}$

${\displaystyle D(x,K):=\min\{d(x,k)\mid k\in K\}.}$

Then one defines the Hausdorff distance between two non-empty compact subsets and as ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \delta (A,B):=\max\{\max\{D(a,B)\mid a\in A\},\max\{D(b,A)\mid b\in B\}\}.}$

One can show that there is indeed a metric on the set of all compact subsets of . ${\displaystyle \delta }$ ${\displaystyle E}$

Equivalently, the Hausdorff distance can be defined as

${\displaystyle \delta (A,B)=\inf\{\epsilon \geq 0\,|\ A\subseteq B_{\epsilon }{\text{ und }}B\subseteq A_{\epsilon }\}}$,

in which

${\displaystyle A_{\epsilon }:=\bigcup _{a\in A}\{z\in E\,|\ d(z,a)\leq \epsilon \}}$,

this is the set of all points with a distance of at most from the set . ${\displaystyle \epsilon }$ ${\displaystyle A}$

Applications

In the theory of iterated function systems , fractals are generated as sequence limit values in the sense of the Hausdorff metric.

See also

• Hausdorff convergence
• Blaschke's selection sentence
• Gromov-Hausdorff metric

literature

• MI Voitsekhovskii: Hausdorff metric . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ James Munkres: Topology . Prentice Hall, 1999, ISBN 0-13-181629-2 , pp. 280-281 ( google.com ).