Answers to questions of Grünbaum and Loewner

Abstract

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