On the unique determination of compact convex sets from their projections. The complex case (Q1307184)

According to a theorem of Hadwiger, two convex bodies in an \(n\)-dimensional (real) Euclidean space \(E^n\), \(n \geq 3\), are directly homothetic if their projections onto each \(k\)-dimensional subspace, \(2\leq k\leq n-1\), are directly homothetic. In the article under review, the author studies a similar problem in the \(n\)-dimensional complex vector space \(\mathbb{C}^n\). A typical result is as follows: Let \(V_1\) and \(V_2\) be compact convex bodies in \(\mathbb{C}^n\), \(n\geq 3\), such that (i) for each 2-dimensional subspace \(P\subset \mathbb{C}^n\), projections of \(V_1\) and \(V_2\) into \(P\) are mutually \(SU(2)\)-congruent and have no nontrivial \(SU(2)\) symmetries; (ii) for each \(i=1,2\) and for every distinct complex lines \(\lambda, \mu\subset C^n\), projections of \(V_i\) into \(\lambda\) and \(\mu\) are not \(U(1)\)-congruent to each other and have no \(U(1)\) symmetries. Then there exists either a parallel displacement or a central symmetry with respect to some point in \(C^n\) which takes \(V_1\) onto \(V_2\).