---
tag: 984ENA3M5Q
kind: definition
name: Sublinear function
msc: [46A22, 52A41]
defines: K3356N0SAX
uses: [000AYPA9AJ]
sources:
  - url: https://en.wikipedia.org/wiki/Sublinear_function
---

## Definition

Let $X$ be a [[vector space|000AYPA9AJ]] over $\R$.
A *sublinear function* on $X$ is a map $p \colon X \to \R$ such that:

1. $p(tx) = t\, p(x)$ for all real $t > 0$ and all $x \in X$ (positive homogeneity).
2. $p(x + y) \le p(x) + p(y)$ for all $x, y \in X$ (subadditivity).

## Remarks

Every norm on $X$ is sublinear.
For example, $p(x) = \abs{x}$ on $\R$ satisfies $\abs{tx} = t\abs{x}$ for $t > 0$ and $\abs{x + y} \le \abs{x} + \abs{y}$.
Every linear functional is sublinear, and so is $p(x_1, x_2) = \max(x_1, x_2)$ on $\R^2$.
The map $p(x) = x^2$ on $\R$ is not sublinear: $p(2 \cdot 1) = 4$ but $2\, p(1) = 2$, so clause 1 fails.
Clause 1 with $t = 2$ and $x = 0$ gives $p(0) = 2p(0)$, so $p(0) = 0$; hence clause 1 also holds for $t = 0$.
A sublinear function $p$ with $p(-x) = p(x)$ for all $x$ is exactly a [[seminorm|FP5GVCN8GV]].