---
tag: 99VGJBCYC0
kind: concept
name: Superadditive function
msc: [39B62, 26A51]
sources:
  - url: https://en.wikipedia.org/wiki/Superadditivity
same_as:
  wikidata: Q922087
  wikipedia: Superadditivity
---

## Summary

A superadditive function is a function $f$ with $f(x + y) \ge f(x) + f(y)$ for all $x, y$ in its domain.
It is the mirror image of a [[subadditive function|3X9DD9J3CV]]: $f$ is superadditive exactly when $-f$ is subadditive.
For superadditive sequences, [[Fekete's lemma|8A4GGX1MEY]] gives that $a_n / n$ converges to its supremum, which may be $+\infty$.
Superadditivity describes situations where joining parts gains something, as for convex functions vanishing at $0$.

## Explanations

### L1 human

A function is superadditive when the whole is worth at least the sum of its parts.
The square function on $[0, \infty)$ is an example: $(3 + 4)^2 = 49$ is more than $3^2 + 4^2 = 25$.
The number of pairs that can be formed from a group of people is another: two groups merged have all the old pairs and some new ones.