We construct a convex body K in dimensions 5 and higher, with the property that there is exactly one hyperplane section of K through its centroid, such that the centroid of this section coincides with the centroid of K. This provides answers to questions of Gruenbaum and Loewner for dimensions starting from 5. The proof is based on the existence of non-intersection bodies in these dimensions. Joint work with S. Myroshnychenko and K. Tatarko.