Geoscience Reference
In-Depth Information
Fig. 7.6
Locating a unit ball of norm k
1
with respect to another norm k
2
.
Left
: The unit circle of
the maximum-norm is to be located, distances are measured w.r.t the rectangular norm.
Right
:The
Euclidean circle is to be located, distances are measured w.r.t the maximum norm
In Körner et al. (
2009
,
2011
) this problem is studied for two
different
norms under
the name
generalized circle location
.
More precisely, given two norms k
1
and k
2
and a set of points V in the plane
with positive weights
w
j
>0, the goal of generalized circle location is to locate and
scale the unit ball of norm k
1
such that the sum of weighted distances between its
circumference and the given points is minimized, where distances are measured by
the other norm k
2
. Figure
7.6
shows two possible situations. In the left part of the
figure, the new facility is the scaled and translated unit circle of the k
1
WD kk
max
norm and the distances to the four given points are measured by the k
2
WD kk
1
norm. In the right part, k
1
WDkk
2
and k
2
WDkk
max
.
In Körner et al. (
2011
), properties of minsum generalized circle location are
investigated, and it is shown that not much of the properties for minsum circle
location still hold. There is neither an easy formula for computing the distance
between a point and such a generalized circle, nor does any of the incidence criteria
hold. In fact, there are examples in which no optimal circle passes through any of
the existing points. However, if both norms k
1
and k
2
are block norms, a finite
dominating set can still be identified (see Körner et al.
2009
). The problem of
locating a general circle is interesting for many special cases, e.g. if a box should be
located. Such cases have been studied in Brimberg et al. (
2011b
).
7.5
Locating Other Types of Dimensional Facilities
7.5.1
Locating Line Segments
The
line segment location problem
looks for a line segment with specified length
which minimizes the distances to the set V of existing points.
