This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \textit{ Let $S$ be an analytic convex sphere-homeomorphic surface in $\mathbb R^3$ and let $k_1(\boldsymbol{ x})\leqslant k_2(\boldsymbol{ x})$ be its principal curvatures at the point $\boldsymbol{ x}$. If the inequalities $k_1(\boldsymbol{ x})\leqslant k\leqslant k_2(\boldsymbol{ x})$ hold true with some constant $k$ for all $\boldsymbol{ x}\in S$ then $S$ is a sphere.} The imphases is on a result of Y. Martinez-Maure who first proved that the above statement is not valid for convex $C^2$-surfaces. For convenience of the reader, in addendum we give a Russian translation of that paper by Y. Martinez-Maure originally published in French in \textit{ C. R. Acad. Sci., Paris, S\'{ e}r. I, Math.} { \bf 332} (2001), 41--44.
Categories: math.DG math.MG