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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $n$. 
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     title = {Estimation of maximal distances between spaces with norms invariant under a~group of operators},
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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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