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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Received December 4, 1996, and in revised form April 14, 1997.
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Gardner, R., Soranzo, A. & Volčič, A. On the Determination of Star and Convex Bodies by Section Functions . Discrete Comput Geom 21, 69–85 (1999). https://doi.org/10.1007/PL00009411
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DOI: https://doi.org/10.1007/PL00009411