Endpoint regularity of 2d Mumford-Shah minimizers

We discuss an epsilon-regularity result at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers obtained in a joint work with C. De Lellis (U. Zuerich). As an outcome of our analysis, if in a ball $B_r(x)$ the jump set of a given Mumford-Shah minimizer is sufficiently close in the Hausdorff distance to a radius of $B_r(x)$, then in a smaller ball the jump set is a connected arc terminating at some interior point and $C^{1,\alpha}$ up to the tip.