Measure properties of the set of initial data yielding non uniqueness for a class of differential inclusions

We study uniqueness property for the Cauchy problem $x'\in\partial V(x)$, $x(0)=\xi$, where $V\colon\mathbb{R}^n\to\mathbb{R}$ is a locally Lipschitz continuous, quasiconvex function (i.e. the sublevel sets $\{V\le c\}$ are convex) and $\partial V(x)$ is the generalized gradient of $V$ at $x$. We prove that if $0\not\in\partial V(x)$ for $V(x)\ge b$, then the set of initial data $\xi\in\{V=b\}$ yielding non uniqueness of solution in a geometric sense has $(n-1)$-dimensional Hausdorff measure zero in $ {V=b\}$.