cone

Examples

1. In $\sR$ , the set $x>0$ is a blunt cone.
2. In $\sR$ , the set $x\ge 0$ is a pointed salient cone.
3. Suppose $x\in \sR^n$ . Then for any $\varepsilon>0$ , the set $$ C=\bigcup \{\, \lambda B_x(\varepsilon) \mid \lambda >0 \,\} $$ is an open cone. If $|x| < \varepsilon$ , then $C=\sR^n$ . Here, $B_x(\varepsilon)$ is the open ball at $x$ with radius $\varepsilon$ .
4. In a normed vector space, a blunt cone $C$ is completely determined 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.

Bibliography

1

M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.

2

J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company, 1966.

3

R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.

4

I.M. Glazman, Ju.I. Ljubic, Finite-Dimensional Linear Analysis, A systematic Presentation in Problem Form, Dover Publications, 2006.