AI 中文总结
该研究建立了Minkowski不对称性的紧稳定性估计，肯定了Belloni和Freund的猜想，并将其应用于改进Banach-Mazur紧集直径的上界。
AI 中文摘要
我们建立了Minkowski不对称性在其最大值附近的紧稳定性估计，改进了Böröczky、Guo和Schneider此前的估计。更确切地说，若n维凸体K的Minkowski不对称性s(K)≥n−ε，其中ε∈[0,1)，则其与n-单纯形的Banach-Mazur距离至多为1+ε+ε²/[2(1−ε)]。该与维数无关的估计在ε的线性阶范围内是最优的。作为应用，我们证明了与欧氏球的最大Banach-Mazur距离的稳定性结果，将Kobos此前的估计改进至最优线性阶：若K与球的Banach-Mazur距离至少为n−ε，其中ε∈[0,1/2)，则其与n-单纯形的Banach-Mazur距离至多为1+2ε+2ε²/(1−2ε)。一个关键要素是对Belloni和Freund猜想的肯定解答，表明每个凸体K都包含其体积最小的外接椭球经因子1/√[n s(K)]缩放后的平移副本。最后，我们将Minkowski不对称性的稳定性估计应用于获得固定维数下Banach-Mazur紧集直径的改进上界。
英文摘要
We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is sharp to the linear order in $\varepsilon$, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor $\sqrt{n s(K)}$. Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the $L_p$-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the $L_p$-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.
Comments31 pages
