# 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 $$\delta_{x_0} f$$ of a Lipschitz map $$f$$, due to F.H.Clarke in On the inverse function theorem:

Theorem. Let $$f : U \subset \mathbb{R}^n \to \mathbb{R}^n$$ be Lipschitz, and suppose that every matrix $$A ∈ \delta_{x_0}f$$ 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 $$f \circ g = \mathbb{1}_V$$ and $$g \circ f = \mathbb{1}_U$$.

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 The inverse function theorem for Lipschitz maps. In the case of $$B_r$$ ball of radius $$r$$ centered in zero his result reads:

Theorem. Let $$f : \overline{B_r} \subset \mathbb{R}^n \to \mathbb{R}^n$$, $$f(O)=O$$, $$L:\mathbb{R}^n \to \mathbb{R}^n$$ invertible linear map and $$\rho<1$$ such that $$\forall x_1,x_2\in \overline{B_r},\quad \lVert L^{-1}(f(x_2)-f(x_1)) - (x_2-x_1) \rVert \leq \rho \lVert x_2 - x_1 \rVert\,.$$ Then $$f$$ is injective on $$\overline{B_r}$$ with $$\forall x_1,x_2\in \overline{B_r},\quad \frac{1-\rho}{\lVert L^{-1}\rVert} \lVert x_2-x_1\rVert \leq \lVert f(x_2)-f(x_1) \rVert \leq \lVert L\rVert (1+\rho) \lVert x_2-x_1 \rVert \,.$$ 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 $$r_1=\rho\frac{1-\rho}{\lVert L^{-1}\rVert}$$.

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, Convergence of Newton's method and inverse function theorem in Banach space.