The Mumford–Shah conjecture on planar minimizer structure

Let $(K,u)$ be a normalized minimizer of the Mumford–Shah functional $E_\lambda$ in a planar domain. The singular set $K$ is the union of finitely many closed $C^1$ arcs $\gamma_i$ that do not cross and may meet at their endpoints at angles of $120$ degrees in triple junctions. In particular, for every $x\in K$, the set $K\cap B_r(x)$ in any sufficiently small disk is diffeomorphic to one of the following: a radius of $B_r(x)$, a diameter of $B_r(x)$, or three radii of $B_r(x)$ meeting at angles of $120$ degrees.

Mumford–Shah conjecture. If $(K,u)$ is a minimizer of $E_\lambda$, then $K$ has the structure described above.

This is the central planar regularity conjecture for the Mumford–Shah functional and remains unsolved. The statement predicts that minimizer singular sets have only smooth arcs, crack tips, and triple junctions, with no further pathological terminal or accumulation points.