Stability of the Blaschke–Santaló and the affine isoperimetric inequality

A stability version of the Blaschke–Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n⩾3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.