I can't find a metric $\delta$ in $\mathbb{R}^2\setminus\{0\}$ such that be equivalent to euclidean metric, be equal to euclidean metric in the unitary circle and for all $r>0$ the set $\{(x,y)\in\mathbb{R}^2\setminus\{0\}; 0
I tried defining $\delta$ by cases, but is really dificult to obtain a metric in this form.