均宽和ℓ-范数的加强不等式
Strengthened inequalities for the mean width and the $\ell$-norm
AI总结：
本文加强了关于凸体均宽和ℓ-范数的不等式，证明了正则单纯形在特定条件下的极值性质，并研究了中心各向同性测度支撑集的凸包的相关稳定性结果。
摘要：
Barthe证明了正则单纯形在John椭球为欧几里得单位球的凸体中最大化均宽；或等价地，在Löwner椭球为欧几里得单位球的凸体中最大化ℓ-范数。Schmuckenschläger验证了逆命题：正则单纯形在Löwner椭球为欧几里得单位球的凸体中最小化均宽。本文证明了这些结果的更强稳定性版本。我们还考虑了中心各向同性测度在单位球上支撑集的凸包的均宽和ℓ-范数的相关稳定性结果。
英文摘要：
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.
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