Answers to questions of Gr\"unbaum and Loewner

Abstract

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