An isoperimetric inequality in the universal cover of the punctured plane
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.
scientific article; zbMATH DE number 5499842