Sections and projections of homothetic convex bodies (Q2490675)

Classical results on the determination of convex bodies from projections or sections are extended to unbounded sets. Let \(K_1,K_2\) be closed convex sets with nonempty interior in Euclidean space \(E^n\), \(n\geq 3\). There exists a line \(l\subset E^n\) with the following property. If the orthogonal projections of \(K_1\) on \(2\)-dimensional planes containing \(l\) are translates of the respective orthogonal projections of \(K_2\), then \(K_1\) is a translate of \(K_2\). Further, let \(p_1\in K_1\), \(p_2\in K_2\). If for every pair of parallel \(2\)-dimensional planes \(L_1\), \(L_2\) through \(p_1\), \(p_2\), respectively, the intersection \(K_1\cap L_1\) is a translate of \(K_2\cap L_2\), then \(K_1\) is a translate of \(K_2\). Corresponding assertions with translations replaced by homotheties are stated as open problems.