This is a joint work with Christos Saroglou and Serhii Myroshnychenko. Let $n\ge 3$ and let \(K\subset\mathbb R^n\) be a convex body. For a two-dimensional linear subspace       \(P\subset\mathbb R^n\), let \(K|P\) be the orthogonal projection of \(K\) onto \(P\).       We prove that if for every two-dimensional subspace \(P\), the planar       convex body \(K|P\) has \(q\)-fold rotational symmetry up to translation, then       for \(q\ge4\), this forces \(K\) to be an Euclidean ball. The case       \(q=3\) is exceptional: non-spherical examples exist.