# Is $A = \{ (z_1 , z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2|\}$ an absorbing, balanced and convex set?

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.

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.

• Why don't you start with providing the definitions you know for "absorbing set", "balanced set," and "convex set". We like to have askers include, in their questions, some sort of context, whether it's what you tried, where and how you encountered the question, orhow you define the terms in use. Here, I think the definitions will lead you to a solution to this question. – Namaste Nov 27 '16 at 18:52
• Ok, I have that $A$ is a balanced set, because, let $w \in \alpha A$ with $w=(\alpha w_1 , \alpha w_2)$ whichs that $(|w_1| \leq |w_2| )$ then, $|\alpha||w_1| \leq |\alpha| |w_2|$, and so $|\alpha w_1| \leq |\alpha w_2 |$.\\ But I don't know how to do with, absorbing and convex :( – Claudia Viviana Orduz Siabato Nov 27 '16 at 19:13
• How does your text define "absorbing" and "convex" with respect to a set? If you're not using a text, review your course notes, lecture notes, etc. You need to know the definitions of the properties you're trying to prove. – Namaste Nov 27 '16 at 19:18
• Add your conclusion and reasoning about A being a balanced set, to your question (edit: insert), and doing so will demonstrate effort on your part. With respect to "absorbing" and "convex sets, it would be sufficient for you to add the definitions of "absorbing set", "convex set"; (edit your question to include those definitions) , and if you need, ask specifically that you need help to evaluate $A$ with respect to how to apply those definitions. – Namaste Nov 27 '16 at 19:25
• Let $X$ a topologycal vector space , a set $A \subset X$ is called \textit{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 \textit{convex} if $tC + (1-t)C \subset C \ \ (0 \leq t \leq 1)$ – Claudia Viviana Orduz Siabato Nov 27 '16 at 19:27