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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.

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    $\begingroup$ I think the question is too broad in its current form but it is certainly closely related to things which are research level mathematics (e.g. uniform convexity, super-reflexivity and the super versions of Radon-Nikodym property and Krein-Milman) $\endgroup$
    – Yemon Choi
    Mar 24 '15 at 14:22
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    $\begingroup$ 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. $\endgroup$ Mar 24 '15 at 16:12
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    $\begingroup$ 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. $\endgroup$ Mar 24 '15 at 16:15
  • $\begingroup$ Alpx, are you calling me "this non-expert". OK, it's true. The question is interesting but too vague. @BillJohnson 's answer is great. I'd like to see it as THE ANSWER (are there some other "THE ANSERWs"? Then let them see them too). $\endgroup$ Mar 24 '15 at 18:31
  • $\begingroup$ OK, I turned it into an answer now that the question has been reopened. $\endgroup$ Mar 24 '15 at 19:50
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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.

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  • $\begingroup$ Thank you. Now, it's easier to find it and possibly make a reference to it within MO. $\endgroup$ Mar 24 '15 at 21:05
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    $\begingroup$ @billjohnson I think there's a factor of 2 missing in the statement here. $\endgroup$
    – Tim Campion
    Mar 24 '15 at 22:30
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This is not exactly an answer, and not exactly giving a condition equivalent to reflexivity, but I want to give a geometric example warning for the development of the geometric intuition.

Let me start from a well-known characterization that a Banach space $X$ is super-reflexive if and only if $X$ can be equivalently renormed with a uniformly convex norm. If you are not familiar with these definitions, please check Wikipedia. Intuitively uniform convexity says that the ball is "uniformly" round. One has that if $x, y$ belong to the unit sphere of $X$ and are $\varepsilon>0$ apart (i.e. $\|x-y\|>\varepsilon$), then the midpoint has to be inside the unit ball, and not on the sphere, and it has to be uniformly "deep", i.e. $$\|\frac{x+y}2\|\le 1- \delta_X(\varepsilon),$$ where $\delta_X(\varepsilon)>0$ and depends only on $\varepsilon$.

Note that the condition is that $X$ can be renormed to satisfy this condition, not that every norm satisfies it. In fact, it is possible to equivalently renorm the Hilbert space $\ell_2$ to have a positive face of the unit sphere of $\ell_1$ (that is a very "flat" set) inside the positive face of renormed $\ell_2$, which is "the most reflexive space". In fact this is possible in every infinite dimensional Banach space.

To see this, let $(x_i, x_i^*)$ in $X\times X^*$ be a biorthogonal system with $\|x_i\| = 1$ and $\|x_i^*\|\leq 2$. (Such a biorthogonal system exists by applying a theorem of Ovsepian and Pelczynski, see for example the book J. Diestel, Sequences and Series in Banach Spaces,page 56, to a separable subspace of $X$ and then extending to functionals on all of $X$ via the Hahn-Banach theorem.)

Then let $$|||x||| = \max\{ |x_1^*(x)|, \frac12 \|x\|, \displaystyle\sup_{i\ne j; \, i, j\geq 2} (\,|x_i^*(x)| + |x_j^*(x)|\, ) \}.$$ This defines an equivalent norm on $X$ with $||| x_1 + x_n|||=1$ and for all $\alpha_n\geq 0$ we have $$|||\, \sum_{n=1}^\infty \alpha_n (x_1 + x_n)\,||| = \sum_{n=1}^\infty \alpha_n.$$

I first learned about this fact from Vitali Milman. I think it goes back to Ptak. The above norm was defined by A. Pelczynski and M. Wojciechowski.

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Several geometric properties equivalent to non-reflexivity for a Banach space were studied by R.C. James in "Some self-dual properties of normed linear spaces". Ann. of Math. Studies 69 (1972), 159-175. See also Section 10 of D. van Dulst's book "Reflexive and superreflexive Banach spaces". Math. Centre Tracts 102. Amsterdam 1978. One of them:

A Banach space $X$ is non-reflexive if and only if there exist $\varepsilon>0$ and a sequence $(x_n)$ in the unit ball of $X$ such that for each $k\in N$, $$ dist(co\{x_1,\ldots,x_k\}, co\{x_{k+1},\ldots\})\geq \varepsilon. $$

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