A note on propagation of singularities of semiconcave functions of two variables

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ${\mathbb{R}}^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\psi (x)=(x,y_1(x)-y_2(x)),x\in [0,\alpha ]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,\alpha ]$. In other words: singularities propagate along arcs with finite turn.