---
tag: WD3RJZC5A7
kind: concept
name: Extreme set
msc: [52A05, 46A55]
sources:
  - url: https://en.wikipedia.org/wiki/Extreme_point
---

## Summary

An extreme set of a set $A$ is a subset $B$ of $A$ with the following property.
Whenever a point of $B$ lies strictly between two points of $A$, both of those points lie in $B$.
It generalises the notion of [[extreme point|3MSGFJM2NX]]: a point is extreme exactly when the one-point set it forms is an extreme set.
The convex extreme subsets of a convex set are its [[faces|KY7SB854VH]].

## Explanations

### L1 human

Think of a filled square.
An edge of the square is an extreme set.
If a point of the edge is the middle of a segment inside the square, the whole segment must run along that edge.
A corner is an extreme set with one point.
The diagonal is not an extreme set, because its centre is also the middle of the other diagonal, which leaves it.