Let K and L be two convex bodies in \({\mathbb {R}}^4\) , such that their orthogonal projections onto every 3-dimensional subspace are rotations of each other or reflections with respect to a 2-dimensional plane in the subspace. Assume also that the complex lines (we identify \({\mathbb {R}^4}\) with \({\mathbb {C}^2}\) ) are the invariant subspaces of the corresponding rotations or reflections. Then \(K=L\) . A similar statement is proved for sections of K and L.
