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X
wPar
f
1
;:::;f
Q
, there exists
z
2

Proof
If x
62

R

2
such that f
q
.
z
/<f
q
.x/

for each q
2

Q

, that means,

Q

\

L
<
.f
q
;f
q
.x//:

z
2

qD1

Hence, we obtain that

Q

\

L
<
.f
q
;f
q
.x//
¤;
:

qD1

Since the implications above can be reversed the proof is concluded. The

remaining results can be proved analogously.

Remark 9.1
For the case Q
D
2 the previous result states that the set

X
wPar
.f
1
;f
2
/ coincides with tangential cusps between the level curves of

functions f
1
.
/ and f
2
.
/ union with

X
.f
2
/ (see Example
9.1
).

Corollary 9.1
If
f
1
;:::;f
Q
are strictly convex functions then

X
.f
1
/
[

X
Par
f
1
;:::;f
Q
D

X
sPar
f
1
;:::;f
Q
:

X
wPar
.f
1
;:::;f
Q
/
D

Example 9.1 (See Fig.
9.1
)
Let us consider the points a
1
D
.0;0/, a
2
D
.8;3/,

a
3
D
.
3;5/ and the functions f
1
.x/
Dk
x
a
1
k
1
, f
2
.x/
Dk
x
a
2
k
1
, f
3
.x/
D

k
x
a
3
k
1
. By Theorem
9.1
,

X
wPar
.f
1
;f
2
/ is the rectilinear thick path joining a
1

X
wPar
.f
1
;f
3
/ is the dark rectangle with a
1
and a
3
as opposite vertices.

In what follows, since we are dealing with general convex, inf-compact func-

tions, we will focus on providing information about the geometrical structure of

X
wPar
.f
1
;f
2
;f
3
/. This characterization will allow us to obtain a geometrical

description of

and a
2
and

X
sPar
f
1
;f
2
;f
3
in the next section for an

important family of functions. Actually, we will characterize

X
Par
f
1
;f
2
;f
3
and

X
wPar
.f
1
;f
2
;f
3
/ as

a kind of hull delimited by the chains of bicriteria solutions of any pair of functions

f
p
, f
q
p;q
D
1;2;3. This result enables us to obtain the set

X
wPar
f
1
;:::;f
Q

by union of three-criteria solution sets already characterized. In order to do that, let

2
/
WD
n
'
j
'
W

k
'.t/
k
2
D1
o
;

0
;

0
!

2
;'continuous; lim

t!1

C
1
.

R

R

R

R

0
;

2
/ is the set of

where
k
x
k
2
is the Euclidean norm of the point x. C
1
.

R

R

0
WD
Œ0;
1
/ into

continuous curves, which map the set of non-negative numbers

R

0
/ is unbounded in

2
.These

curves are introduced to characterize the geometrical locus of the points surrounded

by weak-Pareto and Pareto chains.

2
and whose image '.

the two-dimensional space

R

R

R