Geometric characterizations of centroids of simplices
type: Item
instance of: scholarly article
title: Geometric characterizations of centroids of simplices
zbMATH Open document ID: 1089.52009
author: Steven G. Krantz
author: John E. McCarthy
author: Harold Parks
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: The authors study centroids of different \(k\)-skeletons of simplices in Euclidean \(d\)-space assuming constant density. They prove that the centroid of the full-dimensional simplex \(S\) and of its vertex set coincide, and that the centroids of the other skeleta are generically different. In addition, types of simplices are investigated for which all the centroids for all values of \(k\) coincide. E.g., it is shown that for any \(d= pq-1\), where \(p\) and \(q\) are integers larger than 2, there exists a non-regular \(d\)-simplex for which the centroids of all \(k\)-skeletons, \(1\leq k\leq d\), coincide, and the conjecture is made that the same holds if \(d+1\) is prime and larger than 3. On the other hand it is proved that the centroids for all \(k\) with \(0\leq k\leq d\) coincide iff the sum of the \(k\)-dimensional areas of the \(k\)-faces containing each specific vertex equals \((k+1)/(d+1)\) times the total \(k\)-dimensional area of the \(k\)-skeleton of \(S\). This is applied to verify the existence of a non-regular 4-simplex all whose centroids coincide. Further on, examples are presented for which centroids coincide in pairs, and it is also shown that the centroids do never coincide in triples if the polytopes under consideration are regular pyramids.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52B12
zbMATH DE Number: 5016169
zbMATH Keywords: convex polytope
zbMATH Keywords: centroid
zbMATH Keywords: (\(d\)-dimensional) simplex
zbMATH Keywords: Diophantine equation
zbMATH Keywords: dynamics
zbMATH Keywords: \(k\)-skeleton
zbMATH Keywords: pyramid
zbMATH Keywords: regular simplex
zbMATH Keywords: rigid body
zbMATH Keywords: Lebesgue measure
MaRDI profile type: Publication
Wikidata QID: Q59701042
full work available at URL: j.jmaa.2005.04.046
OpenAlex ID: W2083457851
cites work: Q5635945
cites work: Q5565773
cites work: A Matter of Gravity
cites work: k-Volume in R n and the Generalized Pythagorean Theorem
DOI: 10.1016/J.JMAA.2005.04.046
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description: scientific article; zbMATH DE number 5016169