The convergence of a sequence of iterated polygons: a discrete combinatorial analysis
type: Item
instance of: scholarly article
title: The convergence of a sequence of iterated polygons: a discrete combinatorial analysis
zbMATH Open document ID: 1436.37053
DOI: 10.1007/978-3-030-20016-9_14
author: Maria Mălin
author: Ionel Roventa
author: Mihai Tudor
publication date: 2019-11-28T00:00:00Z
review text: Let \(\lambda\in(0,1)\) be an arbitrarily fixed parameter. One considers the dynamical system \(T\) that associates to each planar polygon \(\mathcal{P}\) (obtained by joining the points \(v_{1},\dots,v_{N},\) in this order) the polygon \(\mathcal{Q}\) whose vertices \(w_{1},\dots,w_{N}\) are defined as follows: \(w_{1}=(1-\lambda)v_{1}+\lambda v_{2},\) \(w_{2}=(1-\lambda)v_{2}+\lambda v_{3},\dots,\) and \(w_{N}=(1-\lambda)v_{N}+\lambda v_{0}.\) It is proved that each trajectory has a limit point, the centroid of the starting polygon. A similar result holds in the context of spaces with global nonpositive curvature (CAT(0) spaces). For the entire collection see [Zbl 1415.39001].
Mathematics Subject Classification ID: 37E15
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 7138187
zbMATH Keywords: iterated polygons
zbMATH Keywords: convexity
zbMATH Keywords: metric space
zbMATH Keywords: barycentre
zbMATH Keywords: geodesic
MaRDI profile type: Publication
full work available at URL: 978-3-030-20016-9 14
OpenAlex ID: W2954893018
description: scientific article; zbMATH DE number 7138187