Barycentric cuts through a convex body

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Barycentric cuts through a convex body

Popis výsledku v původním jazyce

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&apos;s question. It follows from known results that for n &gt;= 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 &gt;= 3.

Název v anglickém jazyce

Barycentric cuts through a convex body

Popis výsledku anglicky

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&apos;s question. It follows from known results that for n &gt;= 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 &gt;= 3.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-04113Y" target="_blank" >GJ19-04113Y: Pokročilé nástroje v kombinatorice, topologii a příbuzných oblastech</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings of the 36th International Symposium on Computational Geometry (SoCG 2020)

ISBN

978-3-95977-143-6

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

16

Strana od-do

1-16

Název nakladatele

Schloss Dagstuhl--Leibniz-Zentrum für Informatik

Místo vydání

Dagstuhl, Germany

Místo konání akce

Curych (online)

Datum konání akce

22. 6. 2020

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

—