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.
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.
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.
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 $k\in N$, $$ dist(co\{x_1,\ldots,x_k\}, co\{x_{k+1},\ldots\})\geq \varepsilon. $$
