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%2F22%3A10453436" target="_blank" >RIV/00216208:11320/22:10453436 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=rM4Yw2b_fx" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=rM4Yw2b_fx</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00454-021-00364-7" target="_blank" >10.1007/s00454-021-00364-7</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 calledbarycentric if p is the barycenter of K INTERSECTION h. In 1961, Grünbaum raised the questionwhether, for every K, there exists an interior point p through which there are at leastn + 1 distinct barycentric hyperplanes. Two years later, this was seemingly resolvedaffirmatively by showing that this is the case if p = p0 is the point of maximaldepth in K. However, while working on a related question, we noticed that one ofthe auxiliary claims in the proof is incorrect. Here, we provide a counterexample; thisre-opens Grünbaum's question. It follows from known results that for n >= 2, there arealways at least three distinct barycentric cuts through the point p0 ELEMENT OF K of maximaldepth. Using tools related to Morse theory we are able to improve this bound: fourdistinct barycentric cuts through p0 are guaranteed if n >= 3.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
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
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete and Computational Geometry
ISSN
0179-5376
e-ISSN
1432-0444
Volume of the periodical
2022
Issue of the periodical within the volume
68
Country of publishing house
US - UNITED STATES
Number of pages
22
Pages from-to
1133-1154
UT code for WoS article
000750681500001
EID of the result in the Scopus database
2-s2.0-85124193143