Strengthened inequalities for the mean width and the $\ell$-norm

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $\ell$-norm of convex bodies whose L\"owner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschl\"ager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of centered isotropic measures on the unit sphere.

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