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 our theorem to arbitrary origin
centered star bodies and to any open connected collection of planes that fills
out ℝn. We have counterexamples to the theorem for every finite order of
osculation. We have similar theorems for the cosine transform and projection
areas.
