ABP inequalities for singular submanifolds of bounded mean curvature

Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation.