---
tag: K3356N0SAX
kind: concept
name: Sublinear function
msc: [46A22, 52A41]
sources:
  - url: https://en.wikipedia.org/wiki/Sublinear_function
same_as:
  wikidata: Q338652
  wikipedia: Sublinear function
---

## Summary

A sublinear function on a real vector space is a real-valued map that is positively homogeneous and subadditive.
Norms, seminorms and linear functionals are all sublinear.
Sublinear functions are the dominating functions in the [[Hahn–Banach theorem|QB5AE3VYZZ]], and the [[Minkowski functional|PAX9SHDHDW]] of a convex set is a basic example.

## Explanations

### L1 human

A sublinear function behaves like a length that may be lopsided.
Scaling a vector by a positive number scales the value by the same number.
The value of a sum is at most the sum of the values, as in the triangle inequality.
On the plane, $p(x_1, x_2) = \max(x_1, x_2)$ is sublinear but is not a norm, because it can be negative.