On the Evolution of Convex Hypersurfaces by the Qk Flow

We prove the existence and uniqueness of a C1, 1 solution of the Qk flow in the viscosity sense for compact convex hypersurfaces Σt embedded in Rn+1 (n ≥ 2). The solution exists up to the time T < ∞ at which the enclosed volume becomes zero. In particular, for compact convex hypersurfaces with flat sides we show that, under a certain non-degeneracy initial condition, the interface separating the flat from the strictly convex side, becomes smooth, and it moves by the Qk-1 flow at least for a short time.