The sharp quantitative isoperimetric inequality

Abstract: A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall. Precisely, we show that the difference between the perimeter of a set and the perimeter of an Euclidean ball with its same volume, bounds from above (in terms of a dimensional constant) the square of the distance (intended as the volume of the symmetric difference) of the set itself from the family of Euclidean balls.