On a Bonnesen Type Inequality Involving the Spherical Deviation

In recent years the stability of the isoperimetric and related inequalities has been the object of many investigations. Roughly speaking, given the well known isoperimetric property of balls, the question is how far a set E ⊂ R is from the unit ball B1 if |E| = |B1| and its perimeter P (E) is close to the perimeter of B1. The first results in this direction where obtained for planar sets by Bernstein in 1905 ([2]) and Bonnesen in 1924 ([3]). In particular in the latter paper it is proved that if E ⊂ R has the same area of the unit disk D and is bounded by a simple closed curve, then there exist two concentric disks Dr1 ⊂ E ⊂ Dr2 of radii r1, r2 such that (r2 − r1) ≤ P 2(E)− P (D) 4π ,