Geoscience Reference
In-Depth Information
d
1
p
1
+ N(B
0
p
1
)
B
0
e
1
B
p
1
(0, 0)
e
4
e
2
d
4
d
2
(0, 0)
p
2
e
3
p
2
+ N(B
0
p
2
)
d
3
Fig. 9.6
Illustration of the unit ball for the `
1
-norm, its dual ball and two normal cones of this dual
ball
a
1
(a
1
+N(B
0
p
1
))
∩
(a
2
+N(B
0
p
2
))
a
2
Fig. 9.7
Illustration of an elementary convex set for the `
1
-norm
(
1985
), a nonempty convex set C is called an elementary convex set if there exists
a family such that C
D
C. If the unit balls are polytopes, then we can obtain
the elementary convex sets as intersections of cones generated by fundamental
directions of these balls pointed at each demand point (for details, see Durier and
Michelot
1985
). The two-dimensional elementary convex sets are called cells. Let
C
denote to the set of these cells. Therefore each cell is a polyhedron whose vertices
are the intersection points, which we denote by
2
there
exists an upper bound on the number of cells which is O..
MG
max
/
2
/ (see Durier and
Michelot
1985
).
In Fig.
9.7
we show an elementary convex set for the `
1
-norm for two points
a
1
, a
2
. In this example the dual norm is the `
1
-norm where its unit ball B
0
has
the extreme points
f
.1;1/;.
1;1/;.
1
;1/;.1;
1/
g
. The normal cones to B
0
at
p
1
D
.1;
1/ and p
2
D
.
1;1/ are given by N.B
0
;p
1
/
D
cone
..1;0/;.0;
1// and
N.B
0
;p
2
/
D
cone
..
1;0/;.0;1//, respectively, where
cone
stands for the conical
hull of its argument. Thus, the elementary convex set C
with
D
.p
1
;p
2
/ is the
rectangle defined by a
1
and a
2
with sides parallel to the coordinates axes.
IP
. Finally, in the case of
R
