Hausdorff metric

Let (X,d) be a metric space, and let X be the family of all closed and boundedPlanetmathPlanetmathPlanetmath subsets of X. Given AX, we will denote by Nr(A) the neighborhoodMathworldPlanetmathPlanetmath of A of radius r, i.e. the set xAB(x,r).

The upper Hausdorff hemimetric is defined by


Analogously, the lower Hausdorff hemimetric is


Finally, the Hausdorff metric is given by


for A,BX.

The following properties follow straight from the definitions:

  1. 1.


  2. 2.

    δ*(A,B)=0 if and only if BA;

  3. 3.

    δ*(A,B)=0 if and only if AB;

  4. 4.

    δ*(A,C)δ*(A,B)+δ*(B,C), and similarly for δ*.

From this it is clear that δ is a metric: the triangle inequality follows from that of δ* and δ*; symmetry follows from δ*(A,B)=δ*(A,B); and δ(A,B)=0 iff both δ*(A,B) and δ*(A,B) are zero iff AB and BA iff A=B.

Hausdorff metric inherits completeness; i.e. if (X,d) is completePlanetmathPlanetmathPlanetmath, then so is (X,δ). Also, if (X,d) is totally boundedPlanetmathPlanetmath, then so is (X,δ).

Intuitively, the Hausdorff hemimetric δ* (resp. δ*) measure how much bigger (resp. smaller) is a set compared to another. This allows us to define hemicontinuity of correspondences.

Title Hausdorff metric
Canonical name HausdorffMetric
Date of creation 2013-03-22 13:28:34
Last modified on 2013-03-22 13:28:34
Owner Koro (127)
Last modified by Koro (127)
Numerical id 11
Author Koro (127)
Entry type Definition
Classification msc 54E35
Synonym Hausdorff distance
Defines Hausdorff hemimetric