A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall. 1. Introduction The classical isoperimetric inequality states that if E is a Borel set in Rn , n ≥ 2, with finite Lebesgue measure |E|, then the ball with the same volume has a lower perimeter, or, equivalently, that () 1/n nωn |E|(n−1)/n ≤ P (E) . Here P (E) denotes the distributional perimeter of E (which coincides with the classical (n − 1)-dimensional measure of ∂E when E has a smooth boundary) and ωn is the measure of the.
