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arXiv · 2404.15188

Answers to questions of Gr\"unbaum and Loewner

S. Myroshnychenko·K. Tatarko·V. Yaskin
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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.

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S. Myroshnychenko, K. Tatarko, V. Yaskin. 2024-04-23. Answers to questions of Gr\"unbaum and Loewner. https://arxiv.org/abs/2404.15188

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