Prove whether $\{ x \in \mathbb R^n \mid a^T_1x \geq b_1 \lor a^T_2x \geq b_2\}$ is convex or not

Question

Prove whether $$\{ x \in \mathbb R^n \mid a^T_1x \geq b_1 \lor a^T_2x \geq b_2\}$$ is convex or not.

I think if $a_1=a_2$, $b_1=b_2$, then it is convex. It's not convex under other situations. Is that correct? Any help to prove that? Thanks a lot!

Answer 1

It is convex for $a_1=a_2$ no matter what $b_1$ and $b_2$ are.

It is not convex in $\mathbb{R}^2$ when $a_1=(1,0)$, $a_2=(0,1)$, and $b_1=b_2=0$ because then your set is $\{(x_1,x_2)\mid x_1\geq 0 \lor x_2\geq 0\}$, which contains $(-2,0)$ and $(0,-2)$ but not the the midpoint $(-1,-1)$.