  This paper concerns the inverse mean curvature flow of convex hypersurfaces
which are Lipschitz in general. After defining a weak solution, we study the
evolution of the singularity by looking at the blow-up tangent cone around each
singular point. We prove the cone also evolves by the inverse mean curvature
flow and each singularity is removed when the evolving cone becomes flat. As a
result, we derive the exact waiting time for a weak solution to be a smooth
solution. In particular, a necessary and sufficient condition for an existence
of smooth classical solution is given.
