Geodesic
Characterizations of almost transitive superreflexive Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 629-636.

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Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space $X$ means that, for every element $u$ in the unit sphere of $X$, we have $$ \limsup _{\Vert h\Vert \rightarrow 0} \frac{\Vert u+h\Vert +\Vert u-h\Vert -2}{\Vert h\Vert}=2. $$ We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.
Classification : 46B04, 46B10, 46B22
Keywords: convex transitive; almost transitive; superreflexive; uniformly smooth; rough norm 
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Guerrero, Julio Becerra; Palacios, Angel Rodriguez. Characterizations of almost transitive superreflexive Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 629-636. http://geodesic.mathdoc.fr/item/CMUC_2001__42_4_a2/