Best constants for the isoperimetric inequality in quantitative form

Abstract

We prove some results on isoperimetric inequalities with quantitative terms. In the 2-dimensional case, our main contribution is a method for determining the optimal coefficients c1,..,cm in the inequality δP.E≥Pm kD1 ck.∑/k=1C.α(αEm) valid for each Borel set E with positive and finite area, with δP(E) and α(E) being, respectively, the isoperimetric deficit and the Fraenkel asymmetry of E. In n dimensions, besides proving existence and regularity properties of minimizers for a wide class of quantitative isoperimetric quotients including the lower semicontinuous extension of δ P.(E) @aplha;(E) 2, we describe a general technique upon which our 2-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in [12]. © European Mathematical Society 2013.