The interior in this case is also a convex cone.

This implies , which means that is a convex cone.

The class of convex cones is also closed under arbitrary linear maps.

The analogous result holds for any topologically open convex cone.

A convex cone is called regular if whenever both and are in the closure .

You can also state theorems like is a convex cone, and use that for formal reasoning.

In fact, it is the intersection of all convex cones containing S plus the origin.

Let C be an open convex cone.

Certainly, b is not in the convex cone a'x+a'x. Hence, there must be a separating hyperplane.

For any nonempty convex set then is a convex cone.

Hence, the hyperplane with normal y indeed separates the convex cone a'x+a'x from b.

Weakly positive and strongly positive forms form convex cones.

A solvency cone is any closed convex cone such that and .

By the spectral theorem, they coincide with the space of squares and form a closed convex cone.

This desired result is achieved through the use of projection operators and two particular important classes of convex cones.

In other words, the perfect half-spaces are the maximal salient convex cones (under the containment order).

A convex cone is said to be proper if its closure, also a cone, contains no subspaces.

It is not difficult to see that is indeed a convex cone in the sense of convex geometry.

The dual cone with respect to the Killing form is the maximal invariant convex cone.

When one is properly contained in the other, there is a continuum of intermediate invariant convex cones.

A convex cone in a vector space with an inner product has a dual cone .

A convex cone is said to be pointed or blunt depending on whether it includes the null vector 0 or not.

Half-spaces (open or closed) are convex cones.

A blunt convex cone is necessarily salient, but the converse is not necessarily true.

It often means a salient and convex cone, or a cone that is contained in an open halfspace of V.