TY - JOUR

T1 - An isoperimetric inequality in the universal cover of the punctured plane

AU - Alon, Noga

AU - Pinchasi, Adi

AU - Pinchasi, Rom

N1 - 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’s research was supported in part by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

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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U2 - 10.1016/j.disc.2007.10.033

DO - 10.1016/j.disc.2007.10.033

M3 - Article

AN - SCOPUS:53049094586

SN - 0012-365X

VL - 308

SP - 5691

EP - 5701

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 23

ER -