Tight Stability Estimates Near the Simplex and Improved Bounds for the Diameter of the Banach-Mazur Compactum in Fixed Dimensions

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving previously known estimates due to Böröczky, Guo, and Schneider. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K)\ge n-\epsilon$ for $\epsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most $$1+\epsilon +\frac{{\epsilon }^2}{2(1-\epsilon )}\mathrm{.}$$ This dimension-independent estimate is best possible up to the linear order in $\epsilon$. As an application, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate of Kobos to the optimal linear order: if $K$ is at Banach-Mazur distance at least $n-\epsilon$ to the ball for $\epsilon \in [0,\frac{1}{2})$, then its Banach-Mazur distance to the $n$-simplex is at most $$1+2\epsilon +\frac{2{\epsilon }^2}{1-2\epsilon }\mathrm{.}$$ A key ingredient is a positive answer to a conjecture by Belloni and Freund, showing that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled by a factor $1\mathrm{/}\sqrt{ns(K)}$. Finally, we apply the stability estimate for the Minkowski asymmetry to obtain improved upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.