Some unsolved problems of geometric tomography (Q2799759)

This paper is an overview of some results and unsolved problems in the following direction of geometric tomography: Let \(V_1\) and \(V_2\) be two convex bodies in the Euclidean space \(\mathbf R^n\), \(n\geq 3\) and let their orthogonal projections \(V_1(P^2)\) and \(V_2(P^2)\) into any plane \(P^2\subset\mathbf R^n\) be superposed in the plane using some (depending on the plane) linear transformation. How different are the bodies \(V_1\) and \(V_2\)? By what transformation of the space they can be converted into each other? Similarly problem on the cross-sections of two convex bodies is formulated.NEWLINENEWLINEA detailed review of the results and unsolved problems of the shape reconstruction of multidimensional convex and starbodies from their projections and cross-sections are contained in the monograph [\textit{R. J. Gardner}, Geometric tomography. 2nd ed. Cambridge: Cambridge University Press (2006; Zbl 1102.52002)].NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].