Hausdorff metric
Let (X,d) be a metric space, and let ℱX be the family of all
closed and bounded subsets of X. Given A∈ℱX, we will denote by
Nr(A) the neighborhood
of A of radius r, i.e. the set
∪x∈AB(x,r).
The upper Hausdorff hemimetric is defined by
Analogously, the lower Hausdorff hemimetric is
Finally, the Hausdorff metric is given by
for .
The following properties follow straight from the definitions:
-
1.
;
-
2.
if and only if ;
-
3.
if and only if ;
-
4.
, and similarly for .
From this it is clear that is a metric: the triangle inequality follows from that of and ; symmetry follows from ; and iff both and are zero iff and iff .
Hausdorff metric inherits completeness; i.e. if is complete, then so is . Also, if is totally bounded
, then so is
.
Intuitively, the Hausdorff hemimetric (resp. ) 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 |