The Hausdorff metric has many applications.
For example, a recent proposal [24]
approved by the United States Army Research Office (ARO) Small Business
Technology Transfer (STTR) Program has as its objective to “Develop software
for terrain representation by smooth (C1-smooth or more) piecewise polynomials
on irregular triangulations using nontraditional metrics.” The proposal explains that terrain is often
represented by “piecewise planar surfaces on triangulated irregular networks
(TINs)”. This method is used because of hardware constraints and because these
models do not exhibit excessive oscillations. One problem with this method is
that TINs create unrealistic flat surfaces within the triangles and non-smooth
transitions or sharp discontinuities when moving from one triangle to another.
It requires a large number of triangles (and a large amount of data) to
minimize these problems. Other methods, for example smooth-surface techniques
such as conventional polynomial and rational splines, radial basis functions,
and wavelets, “require too much data, too much computing time, too much human
interaction and/or do not preserve shape well.” The proposal states that
nontraditional metrics, including the Hausdorff metric, “can increase the
accuracy and/or efficiency of both piecewise planar surfaces and smooth
piecewise polynomials on irregular triangulations.” The United States military
has also used the Hausdorff distance in target recognition procedures [18].

Among the many other areas that have found
applications of the Hausdorff metric are

·
image
matching and visual recognition by robots [9, 10] and the Computer Vision Group
at Cornell University http://www.cs.cornell.edu/vision/hausdorff/hausmatch.html.

·
medicine [13]

·
image
analysis [22, 23]

·
astronomy
[19]

·
roots of
polynomials [21]

·
graph
theory [4].

Other
applications of the Hausdorff metric can be found in [8, 20], and the metric
also plays an important role in fractal geometry [1]. In most applications the Hausdorff distance
is used to compare what is seen with pre-programmed or recognized patterns --
the smaller the distance the better the match. In the remainder of this paper
we investigate the geometry the Hausdorff metric imposes on the space *H*(**R*** ^{n}*).