Geoscience Reference
In-Depth Information
Proposition 10.1
Let
f
.x/;f
.x/
2 OMf
.n/
.
I.
f
.x/
is a continuous function.
II.
f
.x/
is a symmetric function, i.e. for any
x
2
n
f
.x/
D
f
.
sort
n
.x//
.
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III.
f
.x/
is a convex function iff
1
:::
n
.
IV. If
c
1
and
c
2
are constants, then the function
c
1
f
.x/
C
c
2
f
.x/
2 OMf
.n/
.
V. I f
f
f
r
.x/
g
is a set of ordered median functions that pointwise converges to a
function
f
,then
f
2 OMf
.n/
.
VI. If
f
f
r
.x/
g
is a set of ordered median functions, all bounded above in each
point
x
of
n
, then the pointwise maximum (or sup) function defined at each
point
x
is not in general an
OMf
.
VII. Let
p<n
1
and
x
p
D
.x
1
;:::;x
p
/
,
x
np
D
.x
pC1
;:::;x
r
/
.If
f
.x/
2
OMf
.n/
then
f
p
.x
p
/
C
f
np
.x
np
/
R
f
.x/
.
VIII. Every ordered median function
OMf
.n/
is a difference of two positively
homogeneous convex functions and has a representation
X
n
f
.x/
D
1
'
i
.x/;
iD1
where
'
r
.x/
D
min
˚
max
f
x
i
1
;x
i
2
;:::;x
i
r
gj
i
1
<
2
< ::: < i
r
and
i
1
;i
2
;:::;i
r
2f
1;:::;n
g
:
Proof
The proof of (1) can be found in Rosenbaum (
1950
). The proof of (3) and (8)
are in Grzybowski et al. (
2011
). The proofs of items (2) and (4) are straightforward
and therefore are omitted. A proof of (5) and counterexamples for (6) and (7) are
given in Nickel and Puerto (
2005
, Examples 1.1 and 1.2).
In order to continue the analysis of the ordered median function we need to
introduce some notation that will be used in the following. Let
P
.1:::n/ be the
set of all the permutations of the first n natural numbers,
P
.1:::n/
Df
W
is a permutation of 1;:::;n
g
:
(10.2)
We write
D
..1/;:::;.n//.
The next result, that we include for the sake of completeness, is well-known and
its proof can be found in the topic by Hardy et al. (
1952
).
Lemma 10.1
Let
x
D
.x
1
;:::;x
n
/
and
y
D
.y
1
;:::;y
n
/
be two vectors in
n
.
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Suppose that
x
y
,then
x
ord
D
.x
.1/
;:::;x
.n/
/
y
ord
D
.y
.1/
;:::;y
.n/
/
.
To understand the nature of the
OMf
we need a precise characterization. This
will be done in the following two results using the concepts of symmetry and
sublinearity.
n
C
Theorem 10.1
A function
f
defined over
is continuous, symmetric and linear
over
f
x
W
0
x
1
:::
x
n
g
if and only if
f
2 OMf
.n/
.
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