Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter

Let E be a local almost-minimizer of the relative perimeter in the open set R^n. We prove a free-boundary monotonicity inequality for E at a point x, under a geometric property called ``visibility'', that is required to satisfy in a neighborhood of x. Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of E at x, as well as the fact that any blow-up of E at x is necessarily a perimeter-minimizing cone within the tangent cone to at x.