  We study the regularity of the distance function to the boundary of a domain
in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary
of the domain to have corners. We obtain an explicit formula for the second
derivative of these distance functions. Furthermore, we study a generalized
notion of the ridge of a domain, which is the set of singularities of a
distance function to the boundary of the domain. We completely characterize the
ridge by using a generalized notion of curvature.
