A counter-example to a conjectured characterization of the sphere

It has long been conjectured that a closed convex surface of class C-+(2) whose principal curvatures K-1, K-2 satisfy the inequality (K-1 - c)(K-2 - c) less than or equal to 0 with some constant c, must be a sphere. Partial results have been obtained by A.D. Aleksandrov, H.F. Munzner and D. Koutroufiotis. We reformulate the conjecture in terms of hedgehogs and we give a counter-example. Besides, we prove the conjecture for surfaces of constant width and give a new proof for analytic surfaces. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.