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' replacing 'area'.

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