Bonnesenʼs inequality for John domains in Rn

We prove sharp quantitative isoperimetric inequalities for John domains in Rn. We show that the Bonnesen-style inequalities hold true in Rn under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7], and Fuglede (1989) [4] to reduce our problem to symmetric domains.