Estimation of maximal distances between spaces with norms invariant under a group of operators

We consider the class $A_\Gamma$ of $n$-dimensional normed spaces with unit balls of the form: $B_U=\operatorname{conv}\bigcup\limits_{\gamma\in\Gamma}\gamma(B^1_n\cup U(B^1_n))$, where $B^1_n$ is the unit ball of $\ell^1_n$, $\Gamma$ is a finite group of orthogonal operators acting in ${\mathbb R}^n$, and $U$ is a “random” orthogonal transformation.
It is proved that this class contains spaces with a large Banach–Mazur distance between them. If the cardinality of $\Gamma$ is of order $n^c$, it is shown that, in the power scale, the diameter of $A_\Gamma$ in the modified Banach–Mazur distance behaves as the classical diameter and is of the order $n$.