An isoperimetric inequality in the universal cover of the punctured plane

Noga Alon; Adi Pinchasi; Rom Pinchasi

doi:10.1016/j.disc.2007.10.033

An isoperimetric inequality in the universal cover of the punctured plane

Noga Alon
, Adi Pinchasi
, Rom Pinchasi

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We find the largest ε{lunate} (approximately 1.71579) for which any simple closed path α in the universal cover over(R² {set minus} Z², ̃) of R² {set minus} Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L (α) ≥ ε{lunate} A (α), where L (α) is the length of α and A (α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R² {set minus} Z².

Original language	English (US)
Pages (from-to)	5691-5701
Number of pages	11
Journal	Discrete Mathematics
Volume	308
Issue number	23
DOIs	https://doi.org/10.1016/j.disc.2007.10.033
State	Published - Dec 6 2008
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics

Keywords

Isoperimetric inequality
Universal covering space

Access to Document

10.1016/j.disc.2007.10.033

Cite this

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title = "An isoperimetric inequality in the universal cover of the punctured plane",

abstract = "We find the largest ε\{lunate\} (approximately 1.71579) for which any simple closed path α in the universal cover over(R2 \{set minus\} Z2,{\~ }) of R2 \{set minus\} Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L (α) ≥ ε\{lunate\} A (α), where L (α) is the length of α and A (α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2 \{set minus\} Z2.",

keywords = "Isoperimetric inequality, Universal covering space",

author = "Noga Alon and Adi Pinchasi and Rom Pinchasi",

note = "Funding Information: We would like to thank Leonid Polterovich for introducing the problem to us and for many important comments. We also thank Misha Bialy for pointing out important references. Noga Alon{\textquoteright}s research was supported in part by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.",

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AU - Alon, Noga

AU - Pinchasi, Adi

AU - Pinchasi, Rom

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PY - 2008/12/6

Y1 - 2008/12/6

N2 - We find the largest ε{lunate} (approximately 1.71579) for which any simple closed path α in the universal cover over(R2 {set minus} Z2, ̃) of R2 {set minus} Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L (α) ≥ ε{lunate} A (α), where L (α) is the length of α and A (α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2 {set minus} Z2.

AB - We find the largest ε{lunate} (approximately 1.71579) for which any simple closed path α in the universal cover over(R2 {set minus} Z2, ̃) of R2 {set minus} Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L (α) ≥ ε{lunate} A (α), where L (α) is the length of α and A (α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2 {set minus} Z2.

KW - Isoperimetric inequality

KW - Universal covering space

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UR - https://www.scopus.com/pages/publications/53049094586#tab=citedBy

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DO - 10.1016/j.disc.2007.10.033

M3 - Article

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SP - 5691

EP - 5701

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 23

ER -

An isoperimetric inequality in the universal cover of the punctured plane

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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