Answers to questions of Gr\"unbaum and Loewner

We construct a convex body K in Rⁿ, n ≥ 5, with the property that there is exactly one hyperplane H passing through $c(K)$, the centroid of K, such that the centroid of K∩ H coincides with $c(K)$. This provides answers to questions of Gr\"unbaum and Loewner for n≥ 5. The proof is based on the existence of non-intersection bodies in these dimensions.