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.

areresearch level mathematics (e.g. uniform convexity, super-reflexivity and the super versions of Radon-Nikodym property and Krein-Milman) $\endgroup$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$1more comment