Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity.
Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = \text{max}\lbrace \vert x + u_1y \vert, \vert x +u_2 y \vert, \vert x + u_3y \vert \rbrace.$$
I'd like to find an explicit homeomorphism between the unit balls in these norms.
Does anyone have any ideas as to how I might proceed?