Geodesic
On the diameter of the Banach-Mazur set
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 95-100.

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On every subspace of $l_{\infty }(\mathbb N)$ which contains an uncountable $\omega $-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin's Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_{\infty }(\mathbb N)$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.
Classification : 03E50, 46B03, 46B20, 46B26
Keywords: Banach-Mazur diameter; elastic Banach spaces; Martin's Maximum axiom 
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Godefroy, Gilles. On the diameter of the Banach-Mazur set. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 95-100. http://geodesic.mathdoc.fr/item/CMJ_2010__60_1_a6/