Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?
I am new in StackExchange I from Colombia, because I don't write English very well.
Edited from comments:
Ok, I have that $A$ is a balanced set, because, let $w\in αA$ with $w=(αw_1,αw_2)$ such that $(|w_1|≤|w_2|)$ then, $|α||w_1|≤|α||w_2|,$ and so $|αw_1|≤|αw_22|.$
My definitions for "absorbing" and "convex" are:
Let $X$ a topological vector space , a set $A \subset X$ is called absorbing if for all $x \in X$ there is a $\lambda > 0 $ such that $x \in \lambda A$, a set $C \subset X$ s called is called convex if $tC + (1-t)C \subset C \ \ (0 \leq t \leq 1)$.
I'd like to see whether/how these two definitions apply in this case.