Strengthened inequalities for the mean width and the $\ell $-norm of origin symmetric convex bodies

Barthe, Schechtman and Schmuckenschläger proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove close-to-be optimal stronger stability versions of these results, together with their counterparts about the $\ell $-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell $-norm of the convex hull of the support of even isotropic measures on the unit sphere.