An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ϵϵ (approximately 1.71579) for which any simple closed path αα in the universal cover R2∖Z2˜ of R2∖Z2R2∖Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥ϵA(α)L(α)≥ϵA(α), where L(α)L(α) is the length of αα and A(α)A(α) is the area enclosed by αα. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2∖Z2R2∖Z2.