# Article

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Keywords:
convex transitive; almost transitive; superreflexive; uniformly smooth; rough norm
Summary:
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.
References:
[1] Aparicio A., Oca na F., Paya R., Rodriguez A.: A non-smooth extension of Fréchet differentiability of the norm with applications to numerical ranges. Glasgow Math. J. 28 (1986), 121-137. MR 0848419
[2] Becerra J., Rodriguez A.: The geometry of convex transitive Banach spaces. Bull. London Math. Soc. 31 (1999), 323-331. MR 1673411 | Zbl 0921.46006
[3] Bourgin R.D.: Geometric aspects of convex sets with the Radon-Nikodym property. Lecture Notes in Mathematics 993, Springer-Verlag, Berlin, 1983. MR 0704815 | Zbl 0512.46017
[4] Cabello F.: Maximal symmetric norms on Banach spaces. Proc. Roy. Irish Acad. 98A (1998), 121-130. MR 1759425 | Zbl 0941.46008
[5] Day M.M.: Normed linear spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag, Berlin, 1973. MR 0344849 | Zbl 0583.00016
[6] Deville R., Godefroy G., Zizler V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Math. 64, New York. 1993. MR 1211634 | Zbl 0782.46019
[7] Finet C.: Uniform convexity properties of norms on superreflexive Banach spaces. Israel J. Math. 53 (1986), 81-92. MR 0861899
[8] Franchetti C., Paya R.: Banach spaces with strongly subdifferentiable norm. Bolletino U.M.I. 7-B (1993), 45-70. MR 1216708 | Zbl 0779.46021
[9] Giles J.R., Gregory D.A., Sims B.: Characterisation of normed linear spaces with Mazur's intersection property. Bull. Austral. Math. Soc. 18 (1978), 105-123. MR 0493266 | Zbl 0373.46028
[10] Kalton N.J., Wood G.V.: Orthonormal systems in Banach spaces and their applications. Math. Proc. Cambridge Philos. Soc. 79 (1976), 493-510. MR 0402471 | Zbl 0327.46022
[11] Skorik A., Zaidenberg M.: On isometric reflexions in Banach spaces. Math. Physics, Analysis, Geometry 4 (1997), 212-247. MR 1484353

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