On hyperbolic and functional analogues of questions of Grünbaum and Loewner

Abstract

Myroshnychenko, Tatarko, and Yaskin constructed a body $K$ in ${\mathbb{R}}^n$, $n\ge 5$, 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 construction provided answers to questions of Grünbaum and Loewner for $n\ge 5$, which are still open in dimensions $3$ and $4$. We study analogues of these questions in the settings of hyperbolic space ${\mathbb{H}}^n$ and $s$-concave functions on ${\mathbb{R}}^n$.