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.