---
tag: J9HA54ACME
kind: concept
name: Pointed cone
msc: [52A20, 46A55]
specializes: [6VE4KA12VD]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_cone
---

## Summary

A pointed cone is a [[convex cone|6VE4KA12VD]] that contains the origin.
Equivalently, it is a nonempty set closed under sums and under multiplication by nonnegative scalars.
So it behaves like a subspace in which only nonnegative coefficients are allowed.
The word is not used uniformly: some authors call a cone pointed when it contains no line through the origin, a property also called salient.

## Explanations

### L1 human

A pointed cone is a wedge that includes its tip at the origin.
You may add vectors of the cone, stretch them by any factor $t \ge 0$, and shrink them all the way to $0$.
The closed first quadrant of the plane is a pointed cone.
The open first quadrant, without its two boundary half-lines and without the origin, is a convex cone that is not pointed in this sense.