Stability estimates for star bodies in terms of their intersection bodies

The paper deals with the problem of estimating the distance, in radial or H ausdorff metrics, between two centred star bodies of R-d, d greater than or equal to 3, in terms of the distance between the corresponding intersectio n bodies. Stability estimates are obtained, under suitable restrictions on the size o f the bodies, in the following classes: (i) bodies such that the distance f rom the origin of every tangent hyperplane is greater than a fixed positive constant; (ii) bodies whose surface area is less than a fixed positive con stant; (iii) for d = 3, 4, bodies whose intersection body is convex.