A rigidity result for global Mumford-Shah minimizers in dimension three

We study global Mumford-Shah minimizers in ${\mathbb{R}}^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u,K)$ in ${\mathbb{R}}^3$, if $K$ is contained in a half-plane and touching its edge, then it is the half-plane itself. This partially answers to a question of Guy David.