Convexity: Show $p\notin K^{\circ}$

Question

Where: $$ K^{\circ}=\{z\in\mathbb{R}^3 \lvert \forall x\in K : z^tx\leq 0\}$$ And: $$\begin{equation*} K:=\left \{ \begin{pmatrix} x_1\\x_2\\x_3 \end{pmatrix} \in \mathbb{R}^3 \Bigg{|} \ x_3^2 \geq x_1^2+x_2^2, x_3 \geq 0 \right \} \end{equation*}$$ Where: $$q= \begin{pmatrix} p_1\\p_2\\p_3 \end{pmatrix} \in \mathbb{R}^3 \setminus -K \text{ and } p_1^2+p_2^2 \neq 0 \text{ and } p_3 < 0$$

And use this to show that $-K=K^{\circ}$.

How should I start the proof? Any pointers?

Answer 1

Edited one. Hint: If $q \in K^{\circ}$ but $q \notin - K$ then you can strongly separate $q$ and $ -K $ via hyperpaln, and then see what will happen...