Infinitesimal rigidity of convex polyhedra through the second derivative of the Hilbert-Einstein functional
type: Item
instance of: scholarly article
zbMATH Open document ID: 1303.52012
DOI: 10.4153/CJM-2013-031-9
author: Ivan Izmestiev
published in: Canadian Journal of Mathematics
publication date: 2014-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.5066
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6331498
zbMATH Keywords: convex polyhedron
zbMATH Keywords: rigidity
zbMATH Keywords: Hilbert-Einstein functional
zbMATH Keywords: Dehn's theorem
zbMATH Keywords: Minkowski theorem
MaRDI profile type: Publication
arXiv ID: 1105.5066
OpenAlex ID: W2322115086
Wikidata QID: Q125989482
title: Infinitesimal rigidity of convex polyhedra through the second derivative of the Hilbert-Einstein functional
review text: Let \(P\) be a compact convex polyhedron in \(\mathbb R^3\) whose vertices are denoted \(p_i\). For small \(t\), let \(P(t)\) be a deformation of \(P\). This is a polyhedron with the same combinatorics as \(P\) and vertices \(p_i+tq_i\) for a given set of vectors \(q_i\). The deformation is called isometric if the edge lengths of \(P(t)\) remain constant in the first order at \(t=0\).NEWLINENEWLINEThe author provides a new proof of Dehn's theorem stating that every convex polyhedron \(P\) in \(\mathbb R^3\) is infinitesimally rigid in the sense that every isometric deformation of \(P\) is trivial. The proof employs derivatives of the discrete Hilbert-Einstein functional on the space of warped polyhedra with a fixed boundary metric.
reviewed by: Sergiy Merenkov
description: scientific article; zbMATH DE number 6331498