Analytic continuation of convex bodies and Funk's characterization of the sphere

A celebrated theorem of P. Funk, 1916, states that an origin-centered star body in R3 is determined by the areas of its central hyperplane cross-sections. In particular, if all these concurrent sections have the same area then the body must be a ball (its boundary is a sphere). It is natural to try to strengthen the theorem by using a smaller class of planes. It is evident that a lower-dimensional class of hyperplanes, e.g., planes passing through an axis, does not suffice, but a proper open subset of planes appears plausible. The class of planes at a small angle relative to an axis has been considered in the literature. We show that this class does not characterize the body. We then show that if a body is known to osculate a ball centered at the origin to infinite order along one hyperplane through the axis, then the proper open class of planes above does determine whether the body is a ball. We generalize ou