Estimates of maximal distances between spaces whose norms are invariant under a group of operators

We consider the class A_г of n-dimensional normed spaces with unit balls of the form: , where B _n ¹ n is the unit ball of ℓ _n ¹ , Γ is a finite group of orthogonal operators acting in ℝⁿ, 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 Γ is of order n^c, it is shown that, in the power scale, the diameter of A_г in the modified Banach-Mazur distance behaves as the classical diameter and is of order n. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 33–42.