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 $s(K) \geq n-\varepsilon$ 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 best possible up to the linear order in $\varepsilon$.

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 $n - \varepsilon$ to the ball for $\varepsilon \in [0,\frac{1}{2})$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + 2\varepsilon + \frac{2\varepsilon^2}{1-2\varepsilon}. \] 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 $1/\sqrt{n s(K)}$.

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.