Almost spherical convex hypersurfaces in \(\mathbb{R}^4\) (Q5956145)

The following theorem is proved: Let \(M\) be an embedded closed convex hypersurface in \(R^4\) and \(H_k\) the \(k\)-th order mean curvature function of \(M\), \(k=0,1,2,3\). For a given \(\varepsilon\), there exists \(\delta (\varepsilon) >0\) which satisfies the following condition: If, for some \(k,1=0, 1,2,3,k\neq 1\), \(\sup_{x\in M}|{H_k(x) \over H_1(x)}-1 |<\delta (\varepsilon)\) then \(M\) lies between two concentric round spheres of radii \(1-\varepsilon\) and \(1+\varepsilon\).