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 ${\displaystyle (s,S)}$ policy in inventory control theory. The policy is characterized by two numbers s and S, ${\displaystyle S\geq s}$, 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 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 ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ is K-convex if

for any ${\displaystyle u,z\geq 0,}$ and ${\displaystyle b>0}$.

A function ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ is K-convex if

for all ${\displaystyle x\leq y,\lambda \in [0,1]}$, where ${\displaystyle {\bar {\lambda }}=1-\lambda }$.

This definition admits a simple geometric interpretation related to the concept of visibility. Let ${\displaystyle a\geq 0}$. A point ${\displaystyle (x,f(x))}$ is said to be visible from ${\displaystyle (y,f(y)+a)}$ if all intermediate points ${\displaystyle (\lambda x+{\bar {\lambda }}y,f(\lambda x+{\bar {\lambda }}y)),0\leq \lambda \leq 1}$ 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