K-convex functions, first introduced by Scarf,[1] are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the
policy in inventory control theory.
The policy is characterized by two numbers s and 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.
Gallego and Sethi [2] have generalized the concept of K-convexity to higher dimensional Euclidean spaces.
Two equivalent definitions are as follows: Let K be a non-negative real number.
is K-convex if for all
x ≤ y , λ ∈ [ 0 , 1 ]
This definition admits a simple geometric interpretation related to the concept of visibility.
A point
if all intermediate points
y , f ( λ x +
lie below the line segment joining these two points.
Then the geometric characterization of K-convexity can be obtain as: It is sufficient to prove that the above definitions can be transformed to each other.
is K-convex, then it is L-convex for any
is convex, then it is also K-convex for any
is K-convex and
α ≥ 0 , β ≥ 0 ,
g = α
( α
is K-convex and
is a random variable such that
is also K-convex.
is K-convex, restriction of
on any convex set
is K-convex.
is a continuous K-convex function and
, then there exit scalars