Barycentric cuts through a convex body

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Department
Series Title
LIPIcs
Abstract
Let K be a convex body in ℝⁿ (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 ∩ 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=p₀ 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 ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.
Publishing Year
Date Published
2020-06-01
Proceedings Title
36th International Symposium on Computational Geometry
Publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
Volume
164
Article Number
62:1 - 62:16
Conference
SoCG: Symposium on Computational Geometry
Conference Location
Zürich, Switzerland
Conference Date
2020-06-22 – 2020-06-26
ISSN
IST-REx-ID
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Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):
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Access Level
OA Open Access
Date Uploaded
2020-06-23
MD5 Checksum
ce1c9194139a664fb59d1efdfc88eaae


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arXiv 2003.13536

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