Semiconcavity and semiconvexity are key regularity properties for functions with many applications in a broad range of mathematical subjects. The notions of semiconcavity and semiconvexity have been adapted to different geometrical contexts, in particular in sub-Riemannian structures such as Carnot groups, where they turn out to be extremely useful for the study of solutions of degenerate PDEs.

In this talk I will show that, for a suitable class of Carnot groups, the Carnot-Carathéodory distance is semiconcave, in the sense of the group, in the whole space. I will also give some applications to solutions of non-coercive Hamilton-Jacobi equations.

Joint work with Qing Liu and Ye Zhang from OIST (Okinawa, Japan)