We say that $\Omega$ is a star-shaped domain (with respect to the origin) of $\mathbb R ^n$ if : 

$$\Omega  = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\; \text{and}\;\; 
 \partial \Omega  = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\} $$
with $g$ is a continuous, positive function on the unit sphere S.

I showed that there is a $\mathcal C^1$ diffeomorphism between $\Omega$ and the unit ball (Euclidean norm $\left \| . \right \|_{2}$).
$$\begin{array}{ccccc}
\Phi & : & B & \to & \Omega \\
 & & y & \mapsto &  y\;h(\frac{y}{\left \| y \right \|}) \\
\end{array}$$
$\Phi$ have some properties: 

• $\Phi$ is well defined.

• $\Phi(\partial B)=\partial \Omega$.

• $\Phi$ is a bijection.

• $\Phi$ is a smooth function.


**Now I would like to show the existence of a Lipschitzian bijection between this domain $\Omega$ and a cube in $\mathbb R ^n$ (norm $\left \| . \right \|_{\infty}$).**

I appreciate your answers and your help.