Barycentric cuts through a convex body

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^n$ (i.e., a compact convex set with nonempty interior). Given a point $p$ in the interior of $K$, a hyperplane $h$ passing through $p$ is called barycentric if $p$ is the barycenter of $K\cap h$. In 1961, Grünbaum raised the question whether, for every $K$, there exists an interior point $p$ through which there are at least $n+1$ distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if $p=p_0$ is the point of maximal depth in $K$. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum's question. It follows from known results that for $n\ge 2$, there are always at least three distinct barycentric cuts through the point $p_0\in K$ of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through $p_0$ are guaranteed if $n\ge 3$.

19 pages, 7 figures