A logarithmic Gauss curvature flow and the Minkowski problem

Let X-0 be a smooth uniformly convex hypersurface and f a positive smooth f unction in S-n. We study the motion of convex hypersurfaces X(., t) with in itial X(., 0) = theta X-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 theta* > 0 such that i f a theta < <theta>*, they shrink to a point in finite time and, if theta > theta*, they expand to an asymptotic sphere. Finally, when theta = theta*, they converge to a convex hypersurface of which Gauss curvature is given e xplicitly by a function depending on f(x), (C) 2000 Editions scientifiques et medicales Elsevier SAS.