I am looking for a Lipschitz continuous bijection from the (closed or open) unit cube $Q$ to the (closed or open) unit ball $B$ in $\mathbb{R}^n$. I found a continuous bijection via
$$f(x) = \max_i |x_i | \frac{x}{|x|}.$$
Now I am struggling to prove that it is Lipschitz though (which I'm pretty sure it is). Can anyone help?
Explicit Lipschitz continuous bijection from unit cube to unit ball
lipschitz-functionsreal-analysis
Best Answer
Your map $f:\mathbb R^n \to \mathbb R^n$ is $$ f(x) = \frac{\|x\|_\infty}{\|x\|_2} x $$ with inverse $$ f^{-1}(x) = \frac{\|x\|_2}{\|x\|_\infty} x .$$
We first want to show that $f$ is Lipschitz, that is, if $\|x - y\|_\infty \le \epsilon$, then there is a constant $C$, depending only on $n$, such that $\| f(x) - f(y) \|_2 \le C \epsilon$.
Consider the following cases.
This shows $f$ is Lipschitz. Showing $f^{-1}$ is Lipschitz is similar.