Abstract
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.
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References
F. L. Bakharev, “Extremally distant normed spaces with additional restrictions,” Mat. Zametki, 79, 339–352 (2006).
E. D. Gluskin, “Diameter of the Minkovski compactum approximately equals n,” Funkts. Analiz Prilozh., 15, 72–73 (1981).
E. D. Gluskin, “Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces,” Mat. Sb., 136(178), 85–96 (1988).
E. D. Gluskin, “Finite-dimensional analogs of spaces without a basis,” Dokl. Akad. Nauk SSSR, 216, 72–73 (1981).
V. I. Gurarii, M. I. Kadets, and V. I. Matsaev, “Distances between finite-dimensional analogs of the spaces L p,” Mat. Sb., 170(112), 481–489 (1966).
A. I. Khrabrov, “Generalized volume ratios and the Banach-Mazur distance,” Mat. Zametki, 70, 918–926 (2001).
I. Bárány and Z. Füredi, “Approximation of the sphere by polytopes having few vertices,” Proc. Amer. Math. Soc., 102, 651–659 (1988).
P. Enflo, “A counterexample to approximation problem for Banach spaces,” Acta Math., 130, 309–317 (1973).
P. Mankiewicz, “Subspace mixing properties of operators in R n with applications to Gluskin spaces,” Studia Math., 88, 51–67 (1988).
S. J. Szarek, “On the existence and uniqueness of complex structure and spaces with ‘few’ operators,” Trans. Amer. Math. Soc., 293, 339–353 (1986).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 17–32.
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Bakharev, F.L. Generalization of some classical results to the case of the modified Banach-Mazur distance. J Math Sci 141, 1517–1525 (2007). https://doi.org/10.1007/s10958-007-0057-x
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DOI: https://doi.org/10.1007/s10958-007-0057-x