Necessary and Sufficient Condition for a Surface to be a Sphere

Let S be a C 1 -smooth closed connected surface in R 3 , the boundary of the domain D , N = N s be the unit outer normal to S at the point s , P be the normal section of D . A normal section is the intersection of D and the plane containing N . It is proved that if all the normal sections for a fixed N are discs, then S is a sphere. The converse statement is trivial.