Geoscience Reference
In-Depth Information
Remark 9.2
It is worth noting that the region
encl
wPar
f
1
;f
2
;f
3
is well-
gen
X
wPar
f
1
;f
2
;f
3
is connected (see Warburton
1983
).
As an illustration of the above result we present the following example.
gen
defined because the set
X
Example 9.2
Let us consider three points a
1
D
.0;0/, a
2
D
.3;
1/ and a
3
D
.3;3/, and the functions f
1
.
/, f
2
.
/ and f
3
.
/ such that,
L
.f
1
;1/
D
.x
1
;x
2
/
W
x
1
9
1
L
.f
2
;1/
D
˚
.x
1
;x
2
/
W
.x
1
3/
2
C
.x
2
C
1/
2
1
L
.f
3
;1/
D
.x
1
;x
2
/
W
.x
1
3/
2
9
4
C
x
2
1
:
C
.x
2
3/
2
4
We can see that these three functions are convex functions. Therefore by the previ-
ous result we obtain the geometrical characterization of the set
X
wPar
.f
1
;f
2
;f
3
/;
this set is the shadowed region in Fig.
9.2
.
Now we are in the right position to show the main result about the geometrical
structure of
X
wPar
.f
1
;:::;f
Q
/.
a
3
X
w
−
Par
(f
1
f
3
)
X
w
−
Par
(f
2
f
3
)
X
w
−
Par
(f
1
f
2
f
3
)
a
1
X
w
−
Par
(f
1
f
2
)
a
2
Fig. 9.2
Illustration of
Example
9.2
