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. A function is K-convex if
for any and .
A function is K-convex if
for all , where .
This definition admits a simple geometric interpretation related to the concept of visibility. [3] Let . A point is said to be visible from if all intermediate points 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. This can be seen by using the transformation
If is K-convex, then it is L-convex for any . In particular, if is convex, then it is also K-convex for any .
If is K-convex and is L-convex, then for is -convex.
If is K-convex and is a random variable such that for all , then is also K-convex.
If is K-convex, restriction of on any convex set is K-convex.
If is a continuous K-convex function and as , then there exit scalars and with such that
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. A function is K-convex if
for any and .
A function is K-convex if
for all , where .
This definition admits a simple geometric interpretation related to the concept of visibility. [3] Let . A point is said to be visible from if all intermediate points 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. This can be seen by using the transformation
If is K-convex, then it is L-convex for any . In particular, if is convex, then it is also K-convex for any .
If is K-convex and is L-convex, then for is -convex.
If is K-convex and is a random variable such that for all , then is also K-convex.
If is K-convex, restriction of on any convex set is K-convex.
If is a continuous K-convex function and as , then there exit scalars and with such that