Namespaces
Variants
Actions

Difference between revisions of "Hausdorff metric"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''Hausdorff distance''
 
''Hausdorff distance''
  
A metric in the space of subsets of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467101.png" />, defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467102.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467103.png" /> be the set of all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467105.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467108.png" /> is a metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h0467109.png" />. Then the Hausdorff metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h04671010.png" /> is the least upper bound of the numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h04671011.png" />. It was introduced by F. Hausdorff in 1914 (see [[#References|[1]]]); one of his most important results is as follows: The space of closed subsets of a compact set is also compact (P.S. Urysohn arrived independently at this theorem in 1921–1922, see [[#References|[2]]]).
+
A metric in the space of subsets of a compact set $K$, defined as follows. Let $X,Y\subset K$ and let $D_{x,y}$ be the set of all numbers $\rho(x,Y)$ and $\rho(y,X)$ where $x\in X$, $y\in Y$ and $\rho$ is a metric in $K$. Then the Hausdorff metric $\operatorname{dist}(X,Y)$ is the least upper bound of the numbers in $D_{x,y}$. It was introduced by F. Hausdorff in 1914 (see [[#References|[1]]]); one of his most important results is as follows: The space of closed subsets of a compact set is also compact (P.S. Urysohn arrived independently at this theorem in 1921–1922, see [[#References|[2]]]).
  
 
====References====
 
====References====
Line 9: Line 10:
  
 
====Comments====
 
====Comments====
Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h04671012.png" />. The Hausdorff metric topology and the [[Exponential topology|exponential topology]] (see also [[Hyperspace|Hyperspace]]) then coincide on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h04671013.png" /> of compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046710/h04671014.png" />.
+
Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space $X$. The Hausdorff metric topology and the [[Exponential topology|exponential topology]] (see also [[Hyperspace|Hyperspace]]) then coincide on the space $K(X)$ of compact subsets of $X$.
  
 
See especially
 
See especially
  
 
in [[Hyperspace|Hyperspace]].
 
in [[Hyperspace|Hyperspace]].

Latest revision as of 14:47, 3 June 2016

Hausdorff distance

A metric in the space of subsets of a compact set $K$, defined as follows. Let $X,Y\subset K$ and let $D_{x,y}$ be the set of all numbers $\rho(x,Y)$ and $\rho(y,X)$ where $x\in X$, $y\in Y$ and $\rho$ is a metric in $K$. Then the Hausdorff metric $\operatorname{dist}(X,Y)$ is the least upper bound of the numbers in $D_{x,y}$. It was introduced by F. Hausdorff in 1914 (see [1]); one of his most important results is as follows: The space of closed subsets of a compact set is also compact (P.S. Urysohn arrived independently at this theorem in 1921–1922, see [2]).

References

[1] F. Hausdorff, "Set theory" , Chelsea, reprint (1978) (Translated from German)
[2] P.S. Urysohn, "Works on topology and other areas of mathematics" , 2 , Moscow-Leningrad (1951) (In Russian)


Comments

Generally, the Hausdorff metric is defined on the space of bounded closed sets of a metric space $X$. The Hausdorff metric topology and the exponential topology (see also Hyperspace) then coincide on the space $K(X)$ of compact subsets of $X$.

See especially

in Hyperspace.

How to Cite This Entry:
Hausdorff metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_metric&oldid=13737
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article