Dual Cones - Adam Oberman Math

Notes from A course in Convexity by Alexander Barvinok

The dual cone is,

$K^* = \{ y \mid \langle y,x \rangle \ge ~0~ \text{ for all } x \in K \}$

note, the sign has changed for the dual cone.

Examples

$K = \{ 2x + y \ge 0 \} \cap \{ x + 2y \ge 0 \}$

$K^* = \{ -x + 2y \ge 0 \} \cap \{ 2x - y \ge 0 \}$

(Figure source: dual cone code)

note the bigger than a quadrant becomes smaller than a quadrant.

For example, the standard norm cone

$K = \{ (x,y,z) \mid z \ge \sqrt{ x^2 + y^2} \}$

can be written as

$K = \{ (x,y,z) \mid \langle v, x \rangle \ge 0, v = (-n_1, -n_2,1) \}$

where $\sqrt{ n_1^2 + n_2^2} = 1$ .

Then

K * = c o n e (v = ( - n 1, - n 2,1))