---
tag: Z5T0SEF3XA
kind: concept
name: Proper cone
msc: [52A20, 90C25]
specializes: [J9HA54ACME]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_cone
---

## Summary

A proper cone in $\R^n$ is a closed [[convex cone|6VE4KA12VD]] with nonempty interior that contains no line.
Proper cones define the generalized inequalities $x \preceq_K y \iff y - x \in K$ used in conic and semidefinite optimisation.
The nonnegative orthant, the second-order cone and the cone of positive semidefinite matrices are the standard examples.
The [[dual cone|QFY1JR1TK4]] of a proper cone is again proper.

## Explanations

### L1 human

A proper cone is a solid, closed wedge with a sharp tip at the origin.
Solid means it has volume, closed means it contains its boundary, and sharp means it does not contain any full line.
The closed first quadrant of the plane is a proper cone.
A closed half-plane is not, because it contains a full line through the origin.