论文标题
Barycentric切入凸体
Barycentric cuts through a convex body
论文作者
论文摘要
令$ k $为$ \ mathbb {r}^n $中的凸形主体（即，一个带有非空内部装置的紧凑型凸形集合）。鉴于$ k $的内部$ p $，如果$ p $是$ k \ cap h $的barycenter，则通过$ p $传递的超平面$ h $称为barycentric。 1961年，格伦鲍姆（Grünbaum）提出了一个问题，即在每$ k $中，都有一个内部点$ p $，至少有$ n+1 $不同的barycentric超平面。两年后，如果$ p = p_0 $是$ k $中最大深度的点，这似乎是肯定地解决的。但是，在提出相关问题时，我们注意到证明中的辅助主张之一是不正确的。在这里，我们提供反例；这个重新打开了格伦鲍姆的问题。从已知的结果来看，对于$ n \ geq 2 $，在最大深度的k $中，总是至少有三个不同的barycentric削减。使用与Morse理论相关的工具，我们能够改善此界限：如果$ n \ geq 3 $，可以保证通过$ P_0 $进行四个不同的Barycentric削减。
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 $n \geq 2$, 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 $n \geq 3$.
