Equivalence of sets

Question

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity

I'm wondering if the following two sets (balls) are equivalent:

$$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 \rbrace$$

$$ \lbrace (v,w) \in \Bbb C^2 : \text{max}\lbrace \vert v+u_1 w\vert,\vert v+u_2 w\vert,\vert v+u_3 w\vert \rbrace \leq 1 \rbrace$$ I haven't yet found a counterexample. If anyone stumbles across a proof or counterexample, I'd be glad to hear of it! Thanks very much.

Answer 1

Let $v=\frac{5}{9}$ and $w=-\frac{5}{9}$. Then $(v,w)$ is clearly not in the first set, but it is in the second since

1) $|v+1w|=0\le1$

2) $|v+(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)w|=\frac{5}{9}|\frac{3}{2}-\frac{\sqrt{3}}{2}i|=\frac{5}{9}\sqrt{3}\le1$

3) $|v+(-\frac{1}{2}-\frac{\sqrt{3}}{2}i)w|=\frac{5}{9}|\frac{3}{2}+\frac{\sqrt{3}}{2}i|=\frac{5}{9}\sqrt{3}\le1$.

Therefore the two sets are not equal, at least.