Suppose $V$ is a vector space (over a field), and $f$ is a function $f\colon V\to\mathbbmss{R}$. Then $f$ is subadditive if

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

## Examples

1. 1.

Any linear function $V\to\mathbbmss{R}$ is subadditive.

2. 2.

If $\|\cdot\|$ is a norm on $V$, $a\geq 0$, then

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

## Properties

Suppose $f$ is subadditive.

1. 1.

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

2. 2.