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

The paper is devoted to generalization of some classical results concerning the Banach-Mazur distance to the modified Banach-Mazur distance. We establish the existense of a space that is uniformly distant in the modified Banach-Mazur distance from all spaces with a small basis constant and of a space that is distant in the modified metric from all spaces admitting a complex structure. The existense of a real space that admits two complex structures and is distant in the sense of the complex modified distance is established. The existense of a space having large generalized volume ratio with all of its subspaces of proportional dimension is proved. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 17–32.