Answers to questions of Gr\"unbaum and Loewner

TL;DR

The paper constructs a convex body in high dimensions with a unique hyperplane through its centroid that preserves the centroid of the intersection, answering longstanding questions in convex geometry.

Contribution

It provides a counterexample in dimensions five and higher, demonstrating the uniqueness of such hyperplanes, based on the existence of non-intersection bodies.

Findings

Existence of a convex body with a unique centroid-preserving hyperplane in dimensions ≥ 5

Answers to questions posed by Gr"unbaum and Loewner in convex geometry

Utilizes non-intersection bodies to establish the main result

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.