The Banach-Mazur distance to the cube in low dimensions (Q542382)

The Banach-Mazur distance between two symmetric convex bodies \(K,L\subset \mathbb{R}^n\) is defined by \(d(K,L):= \inf\{\lambda>0: K\subseteq TL\subseteq \lambda K \; \text{for} \; T\in GL(\mathbb{R}^n)\}\). \textit{M. Lassak} [Bull. Pol. Acad. Sci., Math. 39, No. 3--4, 219--223 (1991; Zbl 0756.52007)] proved the upper bound \(d(K,B^n) \leq \sqrt{n^2-n+1}\) for symmetric convex bodies \(K\) (\(K=-K\)) and the \(n\)-dimensional cube \(B^n=[-1,1]^n\). The main result of the present paper is: \(d(K,B^n)\leq \sqrt{n^2-2n+2 +2/(\sqrt{n+2}-1)}\) for general \(n\); this improves Lassak's upper bound for \(n\geq 3\), but for large \(n\), asymptotically better bounds are known.