© 2015 Académie des sciences. We give an analytic proof of the quantitative isoperimetric inequality in the plane and give an estimation of the upper bound of the constant via maximizing the L∞-norm of the gradient of solutions to the Poisson equation.
Li, G., Zhao, X., Ding, Z., & Jiang, R. (2015). An analytic proof of the planar quantitative isoperimetric inequality. Comptes Rendus Mathematique, 353(7), 589-593. https://doi.org/10.1016/j.crma.2015.04.006