K-convex function

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 $S$, $$S \geq s$$, 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.

Definition
Two equivalent definitions are as follows:

Definition 1 (The original definition)
Let K be a non-negative real number. A function $$g: \mathbb{R}\rightarrow\mathbb{R}$$ is K-convex if
 * $$g(u)+z\left[\frac{g(u)-g(u-b)}{b}\right] \leq g(u+z) + K$$

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

Definition 2 (Definition with geometric interpretation)
A function $$g: \mathbb{R}\rightarrow\mathbb{R}$$ is K-convex if
 * $$g(\lambda x+\bar{\lambda} y) \leq \lambda g(x) + \bar{\lambda} [g(y)+K]$$

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

This definition admits a simple geometric interpretation related to the concept of visibility. Let $$a \geq 0$$. A point $$(x,f(x))$$ is said to be visible from $$(y,f(y)+a)$$ if all intermediate points $$(\lambda x+\bar{\lambda} y, f(\lambda x+\bar{\lambda} y)), 0\leq \lambda \leq 1$$ lie below the line segment joining these two points. Then the geometric characterization of K-convexity can be obtain as:


 * A function $$g$$ is K-convex if and only if $$(x,g(x))$$ is visible from $$(y,g(y)+K)$$ for all $$y\geq x$$.

Proof of Equivalence
It is sufficient to prove that the above definitions can be transformed to each other. This can be seen by using the transformation
 * $$ \lambda = z/(b+z),\quad x=u-b,\quad y=u+z.$$

Property 1
If $$g: \mathbb{R}\rightarrow\mathbb{R}$$ is K-convex, then it is L-convex for any $$L\geq K$$. In particular, if $$g$$ is convex, then it is also K-convex for any $$K\geq 0$$.

Property 2
If $$g_1$$ is K-convex and $$g_2$$ is L-convex, then for $$\alpha \geq 0, \beta \geq 0,\; g=\alpha g_1 +\beta g_2$$ is $$(\alpha K+\beta L)$$-convex.

Property 3
If $$g$$ is K-convex and $$\xi$$ is a random variable such that $$E|g(x-\xi)|<\infty$$ for all $$x$$, then $$Eg(x-\xi)$$ is also K-convex.

Property 4
If $$g: \mathbb{R}\rightarrow\mathbb{R}$$ is K-convex, restriction of $$g$$ on any convex set $$\mathbb{D}\subset\mathbb{R}$$ is K-convex.

Property 5
If $$g: \mathbb{R}\rightarrow\mathbb{R}$$ is a continuous K-convex function and $$g(y)\rightarrow \infty$$ as $$|y|\rightarrow \infty$$, then there exit scalars $$s$$ and $$S$$ with $$s\leq S$$ such that
 * $$g(S)\leq g(y)$$, for all $$y\in \mathbb{R}$$;
 * $$g(S)+K=g(s)<g(y)$$, for all $$y<s$$;
 * $$g(y)$$ is a decreasing function on $$(-\infty, s)$$;
 * $$g(y)\leq g(z)+K$$ for all $$y, z$$ with $$s\leq y\leq z$$.