Dual cone

\textbf{Dual cone}. Let $$K$$ be a cone. Then, the set $$K^* = \{y : x^T y \ge 0 \text{ for all } x \in K\}$$ is called the dual cone of $$K$$. Very interestingly, a dual cone is convex even if the original cone is not.

An intuitive way of grasping the meaning a dual cone is shown in Fig. 369 by Dattoro : Draw the two hyperplanes that are orthogonal to the two vertices that define the original cone $$\mathcal K$$ in the figure, and then their intersection (i.e., the purple area) is the dual cone (the intersection is necessary because the definition of the dual cone says that any vector in the dual cone should form an acute angle with all points of the original cone). As the figure shows, the smaller the original cone is the larger the dual cone. (See also property 4 below).

Some properties of dual cone $$K^*$$ (p51 and p53):
 * 1) A dual cone $$K^*$$ is a ... cone.
 * 2) $$K^*$$ is always convex.
 * 3) $$K^*$$ is always closed.
 * 4) $$K_1 \subseteq K_2 \implies K_2^* \subseteq K_1^*$$.
 * 5) If $$K$$ has nonempty interior, then $$K^*$$ is pointed.
 * 6) If the closure of $$K$$ is pointed, than $$K^*$$ has nonempty interior.
 * 7) $$K^{**}$$ is the closure of the convex hull of $$K$$ (hence, if $$K$$ is convex and closed, $$K^{**}=K$$).
 * 8) If $$K$$ is a proper cone, then $$K^*$$ is also a proper cone.
 * 9) The dual cone of $$K=\mathbf R^n_+$$ is itself.
 * 10) The dual cone of a line in space is its orthogonal complement.
 * 11) More generally, the dual cone of a subspace $$V \subseteq \mathbf R^n$$ is its orthogonal complement $$\{ y : y^Tv  = 0 \text{ for all } v \in V \}$$