Barycentric cuts through a convex body
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10421999" target="_blank" >RIV/00216208:11320/20:10421999 - isvavai.cz</a>
Result on the web
<a href="https://drops.dagstuhl.de/opus/volltexte/2020/12220" target="_blank" >https://drops.dagstuhl.de/opus/volltexte/2020/12220</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2020.62" target="_blank" >10.4230/LIPIcs.SoCG.2020.62</a>
Alternative languages
Result language
angličtina
Original language name
Barycentric cuts through a convex body
Original language description
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 INTERSECTION 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=po 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 po ELEMENT OF K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through po are guaranteed if n >= 3.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GJ19-04113Y" target="_blank" >GJ19-04113Y: Advanced tools in combinatorics, topology and related areas</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
Proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020)
ISBN
978-3-95977-143-6
ISSN
1868-8969
e-ISSN
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Number of pages
16
Pages from-to
1-16
Publisher name
Schloss Dagstuhl--Leibniz-Zentrum für Informatik
Place of publication
Dagstuhl, Germany
Event location
Curych (online)
Event date
Jun 22, 2020
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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