# Cone

### From Glossary

A set, , with the property that if , then , for all positive real . A *convex cone* is a cone that is also a convex set. Equivalently, is a convex cone if
. (An example of a cone that is not convex is the union of the axes.) A *polyhedral cone* is a cone that is also a polyhedron; equivalently, is a polyhedral cone if there exists a matrix such that
. An example of a cone that is not
polyhedral is .

A *quadratic cone* is of the form , where is any (real) matrix. If is negative semi-definite, the cone is all of space. If is positive definite, the cone is just the origin,
. So, quadratic cones usually arise when is indefinite. Example:
. See each of the following special cones: