Mean Convex Smoothing of Mean Convex Cones

We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson (Geom. Meas. Theor. Calcu. Var. 44:441, 1986, Problem 5.7).