Geodesic
Estimation of maximal distances between spaces with norms invariant under a group of operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 33-42 
F. L. Bakharev. Estimation of maximal distances between spaces with norms invariant under a group of operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 33-42. http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a2/
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     author = {F. L. Bakharev},
     title = {Estimation of maximal distances between spaces with norms invariant under a~group of operators},
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     year = {2006},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a2/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We consider the class $A_\Gamma$ of $n$-dimensional normed spaces with unit balls of the form: $B_U=\operatorname{conv}\bigcup\limits_{\gamma\in\Gamma}\gamma(B^1_n\cup U(B^1_n))$, where $B^1_n$ is the unit ball of $\ell^1_n$, $\Gamma$ is a finite group of orthogonal operators acting in ${\mathbb R}^n$, 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 $\Gamma$ is of order $n^c$, it is shown that, in the power scale, the diameter of $A_\Gamma$ in the modified Banach–Mazur distance behaves as the classical diameter and is of the order $n$.

[1] F. L. Bakharev, “Ekstremalno dalekie normirovannye prostranstva s dopolnitelnymi ogranicheniyami”, Mat. Zametki, 79 (2006), 339–352 | MR | Zbl

[2] E. D. Gluskin, “Ekstremalnye svoistva ortogonalnykh parallelepipedov i ikh prilozheniya k geometrii banakhovykh prostranstv”, Mat. sb., 136(178):1(5) (1988), 85–96 | MR | Zbl

[3] A. I. Khrabrov, “Otsenki rasstoyanii mezhdu summami prostranstv $\ell^p_n$”, Vestn. S. -Peterburg. un-ta, Ser. 1, 2000, no. 3(17), 57–63. | MR | Zbl

[4] P. Mankiewicz, N. Tomczak-Jaegermann, “Rotating the unit ball of $\ell^n_1$”, C. R. Acad. Sci. Paris, Serie I, 327 (1998), 167–172 | MR | Zbl

[5] P. Mankiewicz, N. Tomczak-Jaegermann, “Quotients of finite-dimensional Banach spaces; random phenomena”, Handbook on the Geometry of Banach spaces, V. 2, eds. W. B. Johnson, J. Lindnstrauss, Elsevier Science, 2003, 1201–1246 | MR

[6] S. J. Szarek, “On the existence and uniqueness of complex stpucture and spaces witn “few” operators”, Trans. Amer. Math. Soc., 293 (1986), 339–353 | DOI | MR | Zbl

[7] S. J. Szarek, “The finite dimensional basis problem with an appendix on nets of Grassmann mainfolds”, Acta Math., 151 (1983), 153–179 | DOI | MR | Zbl

[8] N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, 38, Longman Scientific Technical, Harlow, New York, 1989 | MR | Zbl