A logarithmic Gauss curvature flow and the Minkowski problem

Let X 0 be a smooth uniformly convex hypersurface and f a postive smooth function in S n .We study the motion of convex hypersurfaces X(•, t) with initial X(•, 0) = θX 0 along its inner normal at a rate equal to log(K/f ) where K is the Gauss curvature of X(•, t).We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ * > 0 such that if θ < θ * , they shrink to a point in finite time and, if θ > θ * , they expand to an asymptotic sphere.Finally, when θ = θ * , they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f (x).