**Question:** If within a convex solid body *C* there is a special point *P* such that every planar section of C passing through P has the same area, then, can we assert that *C* is a sphere and *P* its center? If not, is there any *C* such that there are more than one such *P*'s?

Note 1: Same question can be asked with 'perimeter' (or moment of inertia or any other moment) replacing 'area'. 

Note 2: If we rephrase the question with 'diameter' instead of 'area', the solid *C* could be an oblate spheroid and if we consider least width, it could be a prolate spheroid.