cone

Definition 1.

Suppose $V$ is a real (or complex) vector space with a subset $C$.

1. If $\lambda C\subset C$ for any real $\lambda >0$,then $C$ is called a cone.

2. If the origin belongs to a cone, then the cone is said to be pointed.Otherwise, the cone is blunt.

3. A pointed cone is salient, if it contains no$1$-dimensional vector subspace of $V$.

4. If $C-x_0$ is a cone for some $x_0$ in $V$,then $C$ is a cone with vertex at $x_0$.

5. A convex pointed cone is called a wedge.

6. A proper cone is a convex cone $C$ with vertex at $0$, such that $C\cap (-C)=\{0\}$. A slightly more specific definition of a proper cone is this entry (http://planetmath.org/ProperCone), but it requires the vector space to be topological.

7. A cone $C$ is said to be generating if $V=C-C$. In this case, $V$ is said to be generated by $C$.

Examples

1. In $ℝ$, the set $x>0$ is a blunt cone.

2. In $ℝ$, the set $x\ge 0$ is a pointed salient cone.

3. Suppose $x\in {ℝ}^n$. Then for any $\epsilon >0$, the set

   $$C=\bigcup \{\lambda B_x(\epsilon )\mid \lambda >0\}$$

   is an open cone. If , then $C={ℝ}^n$.Here,$B_x(\epsilon )$ is the open ball at $x$ with radius $\epsilon$.

4. In a normed vector space, a blunt cone $C$ is completelydetermined by the intersection of $C$ with the unit sphere.

Properties

1. The union and intersection of a collection of cones is a cone. In other words, the set of cones forms a complete lattice.

2. The complement of a cone is a cone. This means that the complete lattice of cones is also a complemented lattice.

3. A cone $C$ is convex iff $C+C\subseteq C$.

   Proof.

   If $C$ is convex and $a,b\in C$, then $\frac{1}{2}a,\frac{1}{2}b\in C$, so their sum, being the convex combination of $a,b$, is in $C$, and therefore $a+b=2(\frac{1}{2}a+\frac{1}{2}b)\in C$ also. Conversely, suppose a cone $C$ satisfies $C+C\subseteq C$, and $a,b\in C$. Then $\lambda a,(1-\lambda )b\in C$ for $\lambda >0$ (the case when $\lambda =0$ is obvious). Therefore their sum is also in $C$.∎

4. A cone containing $0$ is a cone with vertex at $0$. As a result, a wedge is a cone with vertex at $0$.

5. The only cones that are subspaces at the same time are wedges.