Polar $C^*$ of the given cone

Question

Derive an explicit form of the polar $C^*$ of the following cone:

$C=\{(x_1,x_2): 0\le x_2\le 2x_1\}$

This is what I've done:

As $C\neq\varnothing, C^*=\{p:p^tx\le 0,\forall x\in C\}$

In this case, $C^*=\{p:p^t(x_1,x_2)\le 0,\forall (x_1,x_2)\in C\}$

To have the equality I think I should write the condition $x_1,x_2\le 0$ but if I do it, then I might be modifiying the $\forall (x_1,x_2)\in C.$

What can I do to solve the problem?

Answer 1

Here is one way of doing it.

Note that for two cones $C_1,C_2$ we have $(C_1 + C_2)^* = C_1^* \cap C_2^*$.

Take $C_1 = \{ \lambda (1,2) \}_{\lambda \ge 0}$ and $C_2 = \{ \lambda (1,0) \}_{\lambda \ge 0}$.

Note that $C_1^* = \{ x | x_1+ 2 x_2 \le 0 \}$, $C_2^* = \{ x | x_1 \le 0 \}$ (both are hyperplanes).

Addendum:

Suppose $R = \{ \lambda p \}_{\lambda \ge 0 }$. Then $R^* = \{ x | \langle x, p \rangle \le 0 \}$.

In $\mathbb{R}^2$, if $p=(p_1,p_2)$ then $R^* = \{ x | x_1 p_1 + x_2 p_2 \le 0 \}$.