The sharp quantitative isoperimetric inequality By N. Fusco, F. Maggi, and A. Pratelli Abstract A quantitative sharp formof 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 R n , n 2, with nite Lebesgue measure jEj, then the ball with the same volume has a lower perimeter, or, equivalently, that (1.1) n! 1=n n jEj (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 !n is the measure of the unit ball B in R n . It is also well known that equality holds in (1.1) if and only if E is a ball. The history of the various proofs and di erent formulations of the isoperi-metric inequality is de nitely a very long and complex one. Therefore we shall not even attempt to sketch it here, but we refer the reader to the many review books and papers (e.g. [3], [18], [5], [21], [7], [13]) available on the subject and to the original paper by De Giorgi [8] (see [9] for the English translation) where (1.1) was proved for the rst time in the general framework of sets of nite perimeter. In this paper we prove a quantitative version of the isoperimetric inequal-ity. Inequalities of this kind have been named by Osserman [19] Bonnesen type inequalities, following the results proved in the plane by Bonnesen in 1924 (see [4] and also [2]). More precisely, Osserman calls in this way any inequality of the form (E) P (E) 2 4jEj ; valid for smooth sets E in the plane R 2 , where the quantity (E) has the following three properties: (i) (E) is nonnegative; (ii) (E) vanishes only when E is a ball; (iii) (E) is a suitable measure of the asymmetry" of E. 942 N. FUSCO, F. MAGGI, AND A. PRATELLI In particular, any Bonnesen inequality implies the isoperimetric inequality as well as the characterization of the equality case. The study of Bonnesen type inequalities in higher dimension has been carried on in recent times in [12], [16], [15]. In order to describe these results let us introduce, for any Borel set E in R n with 0 < jEj < 1, the isoperimetric de cit of E D(E) := P (E) n! 1=n n jEj (n 1)=n 1 = P (E) P (rB ) P (rB ) ; where r is the radius of the ball having the same volume as E, that is jEj = r n jBj. The paper [12] by Fuglede deals with convex sets. Namely, he proves that if E is a convex set having the same volume of the unit ball B then minfH (E; x + B) : x 2 R n g C(n)D(E) (n) ; where H (; ) denotes the Hausdor distance between two sets and (n) is a suitable exponent depending on the dimension n. This result is sharp, in the sense that in [12] examples are given showing that the exponent (n) found in the paper cannot be improved (at least if n 6 = 3). When dealing with general nonconvex sets, we cannot expect the isoperi-metric de cit to control the Hausdor distance from E to a ball. To see this it is enough to take, in any dimension, the union of a large ball and a far away tiny one or, if n 3, a connected set obtained by adding to a ball an arbitrarily long (and suitably thin) entacle". It is then clear that in this case a natural notion of asymmetry is the so-called Fraenkel asymmetry of E, de ned by (E) := min d(E; x + rB ) r n : x 2 R n ; where r > 0 is again such that jEj = r n jBj and d(E; F ) = jEF j denotes the measure of the symmetric di erence between any two Borel sets E; F . This kind of asymmetry has been considered by Hall, Hayman and Weits-man in [16] where it is proved that if E is a smooth open set with a suciently small de cit D(E), then there exists a suitable straight line such that, denoting by E the Steiner symmetral of E with respect to the line (see de nition in Section 3), one has (1.2) (E) C(n) p (E ) : Later on Hall proved in [15] that for any axially symmetric set F (1.3) (F ) C(n) p D(F ) and thus, combining (1.3) (applied with F = E ) with (1.2), he was able to conclude that (1.4) (E) C(n)D(E ) 1=4 C(n)D(E) 1=4 ;
