# radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke:

Teorem. Let $f : U ⊂ \mathbb{R}^n → \mathbb{R}^n$ be Lipschitz, and suppose that every matrix $A ∈ δ_{x_0}f$ is invertible. Then there are neighbourhoods $U$ of $x_0$ and $V$ of $f(x_0)$ in $\mathbb{R}^n$ and a Lipschitz map $g : V → U$ such that $f ◦ g = \mathbb{1}_V$ and $g ◦ f = \mathbb{1}_U$.

I am looking for a reference to an inverse function theorem in the similar setting which provides a bound for the inner radius of the neighbourhoods $U$ and $V$.

The closest I have been able to find is due to Ralph Howard's note. In the case of $B_r$ ball of radius $r$ centered in zero his result reads:

Teorem. Let $f : \overline{B_r} ⊂ \mathbb{R}^n → \mathbb{R}^n$, $f(O)=O$, $L:\mathbb{R}^n → \mathbb{R}^n$ invertible linear map and $\rho<1$ such that $$\forall x_1,x_2\in \overline{B_r},\quad \| L^{-1}(f(x_2)-f(x_1)) - (x_2-x_1) \| \leq \rho \| x_2 - x_1 \|\,.$$ Then $f$ is injective on $\overline{B_r}$ with $$\forall x_1,x_2\in \overline{B_r},\quad \frac{1-\rho}{\|L^{-1}\|} \|x_2-x_1\| \leq \| f(x_2)-f(x_1) \| \leq \|L\| (1+\rho) \| x_2-x_1 \| \,.$$ Furthermore, $f$ restricted to open ball $B_r$ is lipeomorphism onto $V=f[B_r]$, where $V$ contains the open ball $B_{r_1}$ of radius $r_1=\rho\frac{1-\rho}{\|L^{-1}\|}$.

In the case of differentiable $f$, results related to what I am looking for are usually based on Newton's iterates. For example, see Theorem 4.1 of Xinghua.