  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.
