More on functional and quantitative versions of the isoperimetric inequality

This paper deals with the famous isoperimetric inequality. In a first part, we give some new functional form of the isoperimetric inequality, and in a second part, we give a quantitative form with a remainder term involving Wasserstein distance of the classical isopemetric inequality. In both parts, we use optimal transportation. Finally, we use our refined isoperimetric inequality in some classical cases arising in convex geometry.