On the Banach-Mazur Distance in Small Dimensions

We establish some results on the Banach-Mazur distance in small dimensions. Specifically, we determine the Banach-Mazur distance between the cube and its dual (the cross-polytope) in $\mathbb{R}^3$ and $\mathbb{R}^4$. In dimension three this distance is equal to $\frac{9}{5}$, and in dimension four, it is equal to 2. These findings confirm Xue's conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.