Geodesic
Generalization of some classical results to the case of the modified Banach--Mazur distance
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 17-32

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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. 
@article{ZNSL_2006_333_a1,
     author = {F. L. Bakharev},
     title = {Generalization of some classical results to the case of the  modified {Banach--Mazur} distance},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--32},
     publisher = {mathdoc},
     volume = {333},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a1/}
}
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F. L. Bakharev. Generalization of some classical results to the case of the  modified Banach--Mazur distance. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 17-32. http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a1/