Sufficient Condition for Subadditivity

Question

A function $f:\mathbb{R}^n \rightarrow [0,\infty)$ is said to be subadditive if $f(x + y) \leq f(x) + f(y), \quad\forall x,y \in \mathbb{R}^n$.

Is there a sufficient condition for subadditivity in this multi-dimensional case?

In one-dimension, for example, if $g:(0,\infty)\rightarrow [0,\infty)$, a sufficient condition is $x \mapsto g(x)/x$ is nonincreasing.

Edit: The question is a little too general. What I am trying to do is check if a function such as eg. $f(x) = |x|^{1/4 + 1/8(2e^{-|x|}-1)},\quad x\in \mathbb{R}^n$ is subadditive without appealing to the definition, as it doesn't look easy to work with in this case.

If $f$ is linear it is true, if $f$ is positive homogeneous and convex it is true... — copper.hat, Jun 02 '14 at 18:07
For functions $[0,\infty)\to\mathbb R$: Convex and $f(0)=0$ implies subadditive: http://math.stackexchange.com/a/80015/ — Martin Sleziak, Jul 09 '14 at 07:54
Kuczma's book An Introduction to the Theory of Functional Equations and Inequalities has a chapter on subadditive functions. Maybe you can find something useful there. — Martin Sleziak, Jul 09 '14 at 07:57

score 2 · Accepted Answer · answered Jul 09 '14 at 06:23

Your function $f$ is of the form $f(x)=g(|x|)$ where $g$ is increasing. Such a function is subadditive if and only if $g$ is. Necessity is clear; for sufficiency observe
$$f(x+y)=g(|x+y|) \le g(|x|+|y|)\le g(|x|)+g(|y|)=f(x)+f(y)$$ So, the problem reduces to showing that the single-variable function $$x\to x^{-3/4+\exp(-x)/4-1/8}$$ is decreasing on $(0,\infty )$... which is more or less a calculus exercise.

Sufficient Condition for Subadditivity

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