Abstract

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 sufice, 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.

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