---
tag: 3MSGFJM2NX
kind: concept
name: Extreme point
msc: [46A55, 52A07]
sources:
  - url: https://en.wikipedia.org/wiki/Extreme_point
same_as:
  wikidata: Q1385465
  wikipedia: Extreme point
---

## Summary

An extreme point of a convex set is a point of the set that lies strictly inside no segment between two other points of it.
Extreme points are the corners of a convex set.
The [[Krein–Milman theorem|RSRAY0Q68S]] says that a compact convex set in a locally convex space is rebuilt from its extreme points by taking closed convex hulls.
They also matter in linear optimization, where a linear function on a compact convex set in $\R^n$ reaches its maximum at an extreme point.

## Explanations

### L1 human

Think of a convex shape, such as a filled polygon or a disc.
A point is extreme if you cannot place it in the middle of a segment that stays in the shape and has other endpoints.
For a filled square the extreme points are the four corners.
For a filled disc every point of the boundary circle is extreme, and no inner point is.