I want to ask this non-expert question:


What does it mean geometrically for a Banach space to be reflexive?


Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some other theorems may be. But this doesn't give me a geometric intuition so far.



reflexive banach space
There is a beautiful result of Odell and Schlumprecht that gives an answer to this question for separable Banach spaces.Odell, E.(1-TX); Schlumprecht, Th.(1-TXAM) Asymptotic properties of Banach spaces under renormings. (English summary) J. Amer. Math. Soc. 11 (1998), no. 1, 175–188.


A separable Banach space is reflexive iff there is an equivalent norm on the space s.t. whenever $(x\_n)$ is a bounded sequence for which $\lim\_n \lim\_m \| x\_n+x\_m \|= 2\lim\_n \|x\_n\|$, the sequence $(x\_n)$ converges.


