We discuss some recent results related to the homogeneous Dirichlet problem for the infinity Laplace equation with constant source in a bounded domain. We characterize the geometry of domains for which an overdetermined problem admits a viscosity solution. An essential tool is a regularity result for viscosity solutions in convex domains, obtained by the convex envelope method introduced by Alvarez, Lasry and Lions. Based on some recent joint works with Ilaria Fragala' (Politecnico di Milano)