An isoperimetric inequality in the universal cover of the punctured plane
type: Item
instance of: scholarly article
title: An isoperimetric inequality in the universal cover of the punctured plane
zbMATH Open document ID: 1159.52009
author: Noga Alon
author: Adi Pinchasi
author: Rom Pinchasi
published in: Discrete Mathematics
publication date: 2009-01-28T00:00:00Z
review text: The authors find the best possible linear isoperimetric inequality for the universal cover of \(\mathbb{R}^2 \setminus \mathbb{Z}^2\). In particular they prove that if \(\alpha\) is a simple closed curve in the universal cover of \(\mathbb{R}^2 \setminus \mathbb{Z}^2\), then \(L(\alpha) \geq \epsilon A(\alpha)\), where the constant \(\epsilon\) (approximately 1.71579) is the best possible. This is a surprising example for a tight isoperimetric inequality in a base space \(X\) that can be lifted to the same tight isoperimetric inequality in \(\tilde X\) (the universal cover space of \(X\)); this is not the case for many other spaces.
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52C05
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 57M10
zbMATH DE Number: 5499842
zbMATH Keywords: linear isoperimetric inequality
zbMATH Keywords: universal cover space
zbMATH Keywords: integer lattice
MaRDI profile type: Publication
full work available at URL: j.disc.2007.10.033
OpenAlex ID: W2098891244
cites work: Area-Perimeter Relations for Two-Dimensional Lattices
cites work: Q4522426
cites work: Q4039749
cites work: Q3941473
cites work: Two problems concerning the area-perimeter ratio of lattice-point-free regions in the plane
cites work: Detecting and decomposing self-overlapping curves
DOI: 10.1016/J.DISC.2007.10.033
description: scientific article; zbMATH DE number 5499842