On a Bonnesen type inequality involving the spherical deviation

In this paper we investigate the stability of the deviation from being a sphere with respect to the isoperimetric deficit for sets of finite perimeter satisfying a mild regularity property, giving an extension to non convex sets of the classical Bonnesen type result of Fuglede for nearly spherical domains. In particular we prove that if a set of finite perimeter E satisfies an interior cone condition with sufficiently wide angles then we have λH(E)⩽Φ(D(E)), where λH(E) is the deviation from a spherical shape with respect to the Hausdorff distance, D(E) denotes the isoperimetric deficit and Φ is an explicit function vanishing continuously at zero and depending on the dimension