 # Cone

A set, $LaTeX: C$, with the property that if $LaTeX: x \in C$, then $LaTeX: a x \in C$, for all positive real $LaTeX: a$. A convex cone is a cone that is also a convex set. Equivalently, $LaTeX: C$ is a convex cone if $LaTeX: C = C + C$. (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, $LaTeX: C$ is a polyhedral cone if there exists a matrix $LaTeX: A$ such that $LaTeX: C = \{ x : Ax \le 0\}$. An example of a cone that is not polyhedral is $LaTeX: \{(x,y,z): x^2+y^2-z^2=0\}$.
A quadratic cone is of the form $LaTeX: \{x: x^T Q x \le 0\}$, where $LaTeX: Q$ is any (real) matrix. If $LaTeX: Q$ is negative semi-definite, the cone is all of $LaTeX: \mathbb{R}^n$ space. If $LaTeX: Q$ is positive definite, the cone is just the origin, $LaTeX: \{0\}$. So, quadratic cones usually arise when $LaTeX: Q$ is indefinite. Example: $LaTeX: \{x : x \ge 0, \; 2x_1x_2 \ge \|(x_3,...,x_n)||^2\}$. See each of the following special cones: