Almost spherical convex hypersurfaces. (Q2706559)

It is well-known that a closed convex hypersurface in \({\mathbb R}^3\) having constant Gaussian curvature \(K\) or constant mean curvature \(H\) is a sphere. It is also true that a closed convex hypersurface in \({\mathbb R}^3\) having constant ratio of \(H\) to \(K\), \(H/K\), is a sphere. These properties naturally give rise to ask for the case when \(K, H\) or \(H/K\) are close to a constant, whether a given closed convex hypersurface is close to a sphere in some sense. The paper under review gives a more explicit answer for \(H/K\) than known before. The authors prove that given \(\epsilon > 0\), there exists \(\delta = \delta(\epsilon) > 0\) such that if \(M\) is a closed convex hypersurface in \({\mathbb R}^3\) satisfying \(\displaystyle{\big | H/K - 1\big| < \delta(\epsilon)},\) then \(M\) must lie between two spheres of radius \(1- \epsilon\) and \(1 + \epsilon\) with the same center.