Suppose $V$ is a vector space (over a field), and $f$ is a function . Then $f$ is subadditive if $$ f(x+y) \le f(x) + f(y), \quad x,y\in V. $$

Examples

1. Any linear function is subadditive.
2. If $\Vert\cdot \Vert$ is a norm on $V$ , $a\ge 0$ , then $$ f(x) = a+\Vert x \Vert $$ is subadditive.

Properties

1. If $f$ is positively $1$ -homogeneous, then $f$ is convex.
2. The sum of two subadditive functions is subadditive.