Generalization of some classical results to the case of the modified Banach–Mazur distance

The paper is devoted to generalization of some classical results about the Banach–Mazur distance to the modified Banach–Mazur distance. The existense of a space uniformly distant in the modified Banach–Mazur distance from all spaces with small basis constant and a space distant in the modified metric from all spaces admitting complex structure is proved. The existense of a real space admitting two complex structures distant in the sense of the complex modified distance is established. The existense of a space having big generalized volume ratio with all of its subspaces of proportional dimension is shown.