Dual cone

Remarks

The set $C^*$ is an intersection of closed half-spaces through $0$, so it is a closed convex cone that contains $0$. If $C$ is a closed convex cone, then $(C^*)^* = C$.

Example: for $C = \R^n_{\ge 0}$, one has $C^* = \R^n_{\ge 0}$. If $y$ has nonnegative coordinates, then $x \cdot y \ge 0$ for each $x \in C$; conversely, taking $x = e_i$ gives $y_i \ge 0$. A cone with $C^* = C$ is called self-dual; the cone of positive semidefinite matrices with the trace inner product is another self-dual cone.

Non-example: the vector $y = (1, -1)$ is not in the dual of $\R^2_{\ge 0}$, since $(0, 1) \cdot y = -1 < 0$.

The polar cone is $C^\circ = -C^* = \set{y : x \cdot y \le 0 \text{ for all } x \in C}$. In a general real inner product space, or with a pairing between a space and its dual, the same formula defines the dual cone.