Flow by the power of the Gauss curvature

We prove that convex hypersurfaces in ${\mathbb{R}}^{n+1}$ contracting under the flow by any power $\alpha >\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere.