For a Lipschitz function $f: X \rightarrow X$, Clarke's generalized Jacobian at $x$ is defined as the convex hull of the following set:
$$\delta f (x) = \text{convex hull} \left \{\lim_{x_i \rightarrow x} Jf(x_i) : x_i \rightarrow x, f \text{ is differentiable at } x_i \right\}.$$
Clarke's inverse function theorem says that if every matrix in $\delta f(x)$ is invertible, then there is a Lipschitz inverse function $f^{-1}$ on some neighborhood of $x$.
It seems natural to guess that the generalized Jacobian of the inverse function $f^{-1}$ is just the convex hull of the inverses of all the matrices in the generalized Jacobian of $f$. That is,
$$\delta (f^{-1}) (x) = \text{convex hull} \{M^{-1} : M \in \delta f \}.$$
Is this true? If not, are there some special cases where it holds? I can't find it in Clarke's papers or anywhere else.
