Two (new?) variants of convex functions

Question

I find that the following two types of functions are useful to my research.

(i) We know that a function $f: \mathbb{R}_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in \mathbb{R}_+^m$ and all $\lambda_1,\lambda_2\geq 0$ s.t. $\lambda_1+\lambda_2=1$, it holds that \begin{equation}f(\lambda_1{\bf x}+\lambda_2 {\bf y})\leq \lambda_1f({\bf x})+\lambda_2f({\bf y}).\end{equation} The first type of function is to drop the restriction of $\lambda_1+\lambda_2=1$, i.e. to replace convex combination with conic combination.

(ii) The second type of function: $f({\bf 0})=0$, and for all ${\bf x},{\bf y},{\bf z}\in \mathbb{R}_+^m$ such that ${\bf y}\geq {\bf x}$, it holds that \begin{equation}f({\bf y})-f({\bf x})\leq f({\bf y}+{\bf z})-f({\bf x}+{\bf z}).\end{equation}

Could anyone tell me whether they are new or not? Are there any useful references? Many thanks!

Answer 1

1. If you throw in positive homogeneity, then the first class of functions is what is called sublinear, see for instance Proposition 1.1.4 ("Fundamentals of Convex Analysis"; Hiriart-Urruty, Claude Lemaréchal).

2. The second class of functions that you mention is essentially the same as the Wright convex functions, see for instance pg. 223 of A.W. Roberts, D.E. Varberg, "Convex functions", Academic Press, 1973.