On convex bodies with rotationally symmetric planar projections

Abstract

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.