Tomography of convex and star bodies
The authors generalize the notion of a star body and its radial function and define the \(i\)-chord function of a star set for every real \(i\). They apply these notions to solve many problems of geometric tomography which concern sections. For example, they obtain the following interesting results: Corollary 3.5. Let \(1 \leq i \leq n - 1\), and suppose \(L\) is a centered star body in \(E^n\). Then \(L\) is uniquely determined, among all centered star bodies in \(E^n\), by the \(\lambda_i\)-measures of all its sections by \(i\)-dimensional subspaces. Corollary 3.6. The only centered star bodies in \(E^n\) of constant \(i\)- section are \(n\)-dimensional balls. Theorem 4.1. For each \(n \geq 2\) and all real \(i\), there is a nonspherical convex body in \(E^n\) which has the origin as an interior \(i\)- equichordal point. Corollary 4.2. For each \(n \geq 2\) and integer \(i\) with \(1 \leq i \leq n - 1\), there is a nonspherical convex body in \(E^n\), containing the origin in its interior and of constant \(i\)-section. Corollary 5.3. Let \(i\) and \(j\) be distinct real numbers, and suppose that \(L_1\), \(L_2\) are star sets in \(E^n\) with equal \(i\)-chord functions and equal \(j\)-chord functions. If \(L_2\) is centered, then \(L_1 = L_2\). Theorem 6.1. There are noncongruent convex polytopes \(P_1\) and \(P_2\) in \(E^n\), containing the origin in their interiors and with equal \(i\)-chord functions for all real numbers \(i\).
scientific article; zbMATH DE number 694957