TITLE: Compactness of the Unit Ball in a Superreflexive Space
QUESTION [5 upvotes]: The unit ball is compact in the weak topology iff the space is reflexive.  Is there an analogous topology under which the unit ball is compact iff the space is super-reflexive?
(I know a space is super-reflexive iff the unit ball is super weakly compact, but I'm not aware of a topology which makes super weak compactness equivalent to compactness in that topology.)

REPLY [3 votes]: I am guessing the answer is No. Because if you take, for instance, the product of a family of $\ell_p$ spaces for $p>1$ and $p$'s tends to 1, then it won't be superreflexive anymore but the unit ball of the product under the sought topology would be compact. 
Edit: This was a quick late night answer and it doesn't work. By product I had $\ell_2$ direct sum in mind but either way it doesn't solve the problem (see the comments below).