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, Gr\"{u}nbaum 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 Gr\"unbaum's question. It follows from known results that for $n \geq 2$, 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 $n \geq 3$.

中文翻译：

---

重心穿过凸体

令 $K$ 是 $\mathbb{R}^n$ 中的一个凸体（即一个内部非空的紧凸集）。给定 $K$ 内部的一个点 $p$，如果 $p$ 是 $K \cap h$ 的重心，则通过 $p$ 的超平面 $h$ 称为重心。1961 年，Gr\"{u}nbaum 提出了一个问题，对于每个 $K$，是否存在一个内点 $p$，通过它至少有 $n+1$ 个不同的重心超平面。两年后，这是如果 $p=p_0$ 是 $K$ 中的最大深度点，似乎可以肯定地解决了这种情况。但是，在处理相关问题时，我们注意到证明中的一个辅助断言是不正确的。在这里，我们提供一个反例；这重新打开了 Gr\"unbaum 的问题。从已知结果可知，对于 $n \geq 2$，总是至少有三个不同的重心切割通过最大深度的点 $p_0 \in K$。使用与莫尔斯理论相关的工具，我们能够改进这个界限：如果 $n\geq 3$，则保证通过 $p_0$ 的四个不同的重心切割。