Abstract
The i th section function of a star body in $ {\Bbb E}$ n gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin, by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak.
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