We say that $\Omega$ is a Strongly Star Shaped domain (with respect to the origin) of $\mathbb R ^n$ if :
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\cup\{0\}\; \text{and}\;\; \partial \Omega = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\} $$ with $g$ is a continuous, positive function on the unit sphere .
Let's assume that $\Omega $ is a Lipschitz domain (Or a $\mathcal C^1$ doamain) I want to find a Bi-Lipschitz application (with a bounded Jacobian) between $\Omega$ and the unit ball ($B$,$\left \| . \right \|_{2}$).
I tried this one$$\begin{array}{ccccc} \Phi & : & B\setminus \{0\} & \to & \Omega\setminus \{0\} \\ & & y & \mapsto & y\;g(\frac{y}{\left \| y \right \|}) \\ \end{array}$$ But, unluckily $\Phi$ is not Bi-Lipschitz (even if it's a bijection).
I appreciate your answers and your help.