---
tag: 4QXDXYK279
kind: concept
name: Smooth maximum
aliases: [smooth-minimum]
sources:
  - wikipedia: en:45456706@r1317993932
    locator: Smooth maximum
    note: adopted
    mention: wikipedia://en/45456706
    sections: [Summary]
same_as:
  wikidata: Q25098823
  wikipedia: Smooth maximum
inventory: [1TF3TJD9T4]
---

## Summary

In mathematics, a **smooth maximum** of an indexed family *x*<sub>1</sub>, ..., *x*<sub>*n*</sub> of numbers is a smooth approximation to the maximum function $\max(x_1,\ldots,x_n),$ meaning a parametric family of functions $m_\alpha(x_1,\ldots,x_n)$ such that for every $\alpha$, the function $m_\alpha$ is smooth, and the family converges to the maximum function $m_\alpha \to \max$ as $\alpha\to\infty$. The concept of **smooth minimum** is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, $m_\alpha \to \max$ as $\alpha \to \infty$ and $m_\alpha \to \min$ as $\alpha \to -\infty$. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.