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 Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner 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.

中文翻译：

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平均宽度和ℓ-范数的强化不等式

Barthe证明了正则单纯形最大化凸体的平均宽度，其约翰椭球（体中包含的最大体积椭球）是欧几里得单位球；或者等价地，正则单纯形最大化$\ell$-Löwner 椭球（包含物体的最小体积椭球）是欧几里得单位球的凸体的范数。Schmuckenschläger 验证了相反的说法；即，正则单纯形最小化凸体的平均宽度，其 Löwner 椭球是欧几里得单位球。在这里，我们证明了这些结果的更强稳定性版本。我们还考虑了平均宽度和$\ell$-单位球面上的中心各向同性测度的支持的凸包的范数。