Difference between revisions of "Hausdorff metric"

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Let $(X,d)$ be a [[metric space]]. Let
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Let $(X,d)$ be a [[metric space]]. Let $N_d(r,A) = \{x \in X \colon d(x,A) < r\}.$
$$N_d(r,A) = \{x \in X \colon d(x,A) < r\}.$$
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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
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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>
$$H_d(A,B) = \inf \{ r > 0 \colon A \subset N_d(r,B) \mathrm{ \hspace{2pt} and \hspace{2pt}} B \subset N_d(r,A)\}.$$
 
Equivalently, the Hausdorff metric can be defined as
 
$$H_d(A,B) = \max \left\{\sup_{a \in A} \inf_{b \in B} d(a,b), \sup_{b\in B}\inf_{a \in A} d(a,b) \right\}.$$
 
  
==Theorem==
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== Theorem ==
[[The Hausdorff metric is a metric on CL(x)]]<br />
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* [[The Hausdorff metric is a metric on CL(x)]]
  
== See also ==
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== See Also ==
 
* [[Metric space|Metric Spaces]]
 
* [[Metric space|Metric Spaces]]
 
* [[Hausdorff space|Hausdorff Space]]
 
* [[Hausdorff space|Hausdorff Space]]
  
=References=
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== 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
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<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

  1. 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