# Difference between revisions of "Hausdorff metric"

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− | Let $(X,d)$ be a [[metric space]]. Let | + | Let $(X,d)$ be a [[metric space]]. Let $N_d(r,A) = \{x \in X \colon d(x,A) < r\}.$ |

− | + | Let $A,B \in \mathrm{CL}(X)$, the [[hyperspace]] of closed subsets of $X$. The Hausdorff metric induced by $d$ is the function $H_d(A,B) = \inf \{ r > 0 \colon A \subset N_d(r,B) \mathrm{ \hspace{2pt} and \hspace{2pt}} B \subset N_d(r,A)\}.$ | |

− | Let $A,B \in \mathrm{CL}(X)$, the [[hyperspace]] of closed subsets of $X$. The Hausdorff metric induced by $d$ is the function | + | Equivalently, the Hausdorff metric can be defined as $H_d(A,B) = \max \left\{\sup_{a \in A} \inf_{b \in B} d(a,b), \sup_{b\in B}\inf_{a \in A} d(a,b) \right\}.$<ref>Illanes, Alejandro ; Nadler, Sam B., Jr. Hyperspaces. Fundamentals and recent advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. Marcel Dekker, Inc., New York, 1999. xx+512 pp. ISBN: 0-8247-1982-4</ref> |

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− | Equivalently, the Hausdorff metric can be defined as | ||

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− | ==Theorem== | + | == Theorem == |

− | [[The Hausdorff metric is a metric on CL(x)]] | + | * [[The Hausdorff metric is a metric on CL(x)]] |

− | == See | + | == See Also == |

* [[Metric space|Metric Spaces]] | * [[Metric space|Metric Spaces]] | ||

* [[Hausdorff space|Hausdorff Space]] | * [[Hausdorff space|Hausdorff Space]] | ||

− | =References= | + | == References == |

− | + | <references group="references" /> |

## Revision as of 05:24, 1 December 2018

Let $(X,d)$ be a metric space. Let $N_d(r,A) = \{x \in X \colon d(x,A) < r\}.$
Let $A,B \in \mathrm{CL}(X)$, the hyperspace of closed subsets of $X$. The Hausdorff metric induced by $d$ is the function $H_d(A,B) = \inf \{ r > 0 \colon A \subset N_d(r,B) \mathrm{ \hspace{2pt} and \hspace{2pt}} B \subset N_d(r,A)\}.$
Equivalently, the Hausdorff metric can be defined as $H_d(A,B) = \max \left\{\sup_{a \in A} \inf_{b \in B} d(a,b), \sup_{b\in B}\inf_{a \in A} d(a,b) \right\}.$^{[1]}

## Theorem

## See Also

## References

- ↑ Illanes, Alejandro ; Nadler, Sam B., Jr. Hyperspaces. Fundamentals and recent advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. Marcel Dekker, Inc., New York, 1999. xx+512 pp. ISBN: 0-8247-1982-4