---
tag: GCPRFTKS4E
kind: concept
name: Convex independence
msc: [52A20, 52C35]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_position
same_as:
  wikidata: Q17007762
  wikipedia: Convex position
---

## Summary

A family of points is convexly independent if no point of the family lies in the [[convex hull|0FQSDC2EQ2]] of the other points.
A set of points with this property is said to be in convex position.
Then every point of the set is an [[extreme point|3MSGFJM2NX]] of its convex hull.
The notion is the convex analogue of linear and affine independence, and it is central in problems such as the Erdős–Szekeres problem on convex polygons.

## Explanations

### L1 human

Points are in convex position when each of them is a corner of the shape they span.
Stretch a rubber band around the points: every point must touch the band at a corner.
The four corners of a square are convexly independent.
If one adds the center of the square, the new set is not, because the center lies inside the square spanned by the corners.