subadditive

Suppose $V$ is a vector space (over a field), and $f$ is afunction $f:V\to ℝ$. Then $f$ is subadditive if

$$f(x+y)\le f(x)+f(y),x,y\in V.$$

Examples

1. Any linear function $V\to ℝ$ is subadditive.

2. If $\parallel \cdot \parallel$ is a norm on $V$, $a\ge 0$, then

   $$f(x)=a+\parallel x\parallel$$

   is subadditive.

Properties

Suppose $f$ is subadditive.

1. If $f$ is positively $1$-homogeneous, then $f$ is convex.

2. The sum of two subadditive functions is subadditive.