The Banach–Mazur Distance to the Cube in Low Dimensions

Abstract

We show that the Banach–Mazur distance from any centrally symmetric convex body in ℝn to the n-dimensional cube is at most $$\sqrt{n^2-2n+2+\frac{2}{\sqrt{n+2}-1}},$$ which improves previously known estimates for “small” n≥3. (For large n, asymptotically better bounds are known; in the asymmetric case, exact bounds are known.) The proof of our estimate uses an idea of Lassak and the existence of two nearly orthogonal contact points in John’s decomposition of the identity. Our estimate on such contact points is closely connected to a well-known estimate of Gerzon on equiangular systems of lines.