Necessary and Sufficient Condition for a Surface to be a Sphere

Let S be a C1-smooth closed connected surface in R3, the boundary of the domain D, N = Ns 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. Mathematics Subject Classification: 53A05.