文章基本信息

标题：Barycentric Cuts Through a Convex Body
本地全文：下载
作者：Zuzana Pat{'a}kov{'a ; Martin Tancer ; Uli Wagner 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：164
页码：62:1-62:16
DOI：10.4230/LIPIcs.SoCG.2020.62
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Let K be a convex body in â"â¿ (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 â^© h. In 1961, GrÃ¼nbaum 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=pâ, 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 GrÃ¼nbaumâs question. It follows from known results that for n â¥ 2, there are always at least three distinct barycentric cuts through the point pâ, â^^ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through pâ, are guaranteed if n â¥ 3.
关键词：convex body; barycenter; Tukey depth; smooth manifold; critical points