K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the
(
s
,
S
)
policy in inventory control theory. The policy is characterized by two numbers s and S,
S
≥
s
, such that when the inventory level falls below level s, an order is issued for a quantity that brings the inventory up to level S, and nothing is ordered otherwise.
Two equivalent definitions are as follows:
A function
g
:
R
→
R
is K-convex if
g
(
u
)
+
z
[
g
(
u
)
−
g
(
u
−
b
)
b
]
≤
g
(
u
+
z
)
+
K
for any
u
,
z
≥
0
,
and
b
>
0
.
A function
g
:
R
→
R
is K-convex if
g
(
λ
x
+
λ
¯
y
)
≤
λ
g
(
x
)
+
λ
¯
[
g
(
y
)
+
K
]
for all
x
≤
y
,
λ
∈
[
0
,
1
]
, where
λ
¯
=
1
−
λ
.
This definition admits a simple geometric interpretation related to the concept of visibility. Let
a
≥
0
. A point
(
x
,
f
(
x
)
)
is said to be visible from
(
y
,
f
(
y
)
+
a
)
if all intermediate points
(
λ
x
+
λ
¯
y
,
f
(
λ
x
+
λ
¯
y
)
)
,
0
≤
λ
≤
1
lie below the line segment joining these two points. Then the geometric characterization of K-convexity can be obtain as:
A function
g
is
K-convex if and only if
(
x
,
g
(
x
)
)
is visible from
(
y
,
g
(
y
)
+
K
)
for all
y
≥
x
.
It is sufficient to prove that the above definitions can be transformed to each other. This can be seen by using the transformation
λ
=
z
/
(
b
+
z
)
,
x
=
u
−
b
,
y
=
u
+
z
.
If
g
:
R
→
R
is K-convex, then it is L-convex for any
L
≥
K
. In particular, if
g
is convex, then it is also K-convex for any
K
≥
0
.
If
g
1
is K-convex and
g
2
is L-convex, then for
α
≥
0
,
β
≥
0
,
g
=
α
g
1
+
β
g
2
is
(
α
K
+
β
L
)
-convex.
If
g
is K-convex and
ξ
is a random variable such that
E
|
g
(
x
−
ξ
)
|
<
∞
for all
x
, then
E
g
(
x
−
ξ
)
is also K-convex.
If
g
:
R
→
R
is a continuous K-convex function and
g
(
y
)
→
∞
as
|
y
|
→
∞
, then there exit scalars
s
and
S
with
s
≤
S
such that
g
(
S
)
≤
g
(
y
)
, for all
y
∈
R
;
g
(
S
)
+
K
=
g
(
s
)
<
g
(
y
)
, for all
y
<
s
;
g
(
y
)
is a decreasing function on
(
−
∞
,
s
)
;
g
(
y
)
≤
g
(
z
)
+
K
for all
y
,
z
with
s
≤
y
≤
z
.
