On the determination of star and convex bodies by section functions
An important concern in geometric tomography is the question of whether an object in a given class can be distinguished from others in the class by various measurements of its projections on planes or sections by planes. If this is the case, we say that the object is determined in the class by the data. For projections, the natural class of objects to consider is the class \(\mathcal K^n\) of convex bodies in \(\mathbb E^n.\) One natural type of data is the \(i\)th projection function of some \(K\in \mathcal K^n,\) which gives the \(i\)-dimensional volume of the projection of \(K\) on each \(i\)-dimensional subspace. It is remarkable that there is a duality in geometric tomography between results on projections of convex bodies and results on sections of star (rather than convex) bodies. The paper continues the development of the dual Brunn-Minkowski theory initiated by \textit{E. Lutwak} [Pac. J. Math. 58, 531-538 (1975; Zbl 0304.52006); Adv. Math. 71, No. 2, 232-261 (1988; Zbl 0657.52002)]. It is shown that no star body in \(\mathbb E^n\) is determined in \(S^n,\) up to reflection in the origin, by its \(i\)-chord function, \(i>0.\) 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. Suppose that \(K\) is a centered convex body in \(\mathbb E^2\). Let \(\mathcal K^n_o\) be the subclass of \(\mathcal K^n\) whose members contain the origin in their interiors. Then \(K\) is not determined in \(\mathcal K^2_o\) by its \(i\)-chord function and hence not by its section function or \(X\)-ray at the origin.
scientific article; zbMATH DE number 1271010