TITLE: Uniformly Convex spaces
QUESTION [8 upvotes]: My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill).  I'm interested in constructing a uniformly convex Banach space with conditional structure without using interpolation.  The constructions of Ferenczi and Maurey-Rosenthal both use interpolation.  
Using existing methods for constructing spaces with conditional structure I think it is possible to construct a hereditarily indecomposable space whose natural basis statisfies a lower $\ell_2$ estimate on any $n$ disjointly supported blocks vectors supported after the $n^{th}$ position on the basis and an upper $\ell_2$ estimate on all finite block sequences.  The space $X$ is sure to be reflexive and probably doesn't contain $\ell_\infty$ finitely represented. 
I would like to have some way of showing that $X$ is uniformly convex and this is where I'm stuck.  Perhaps one could show that $\ell_1$ is not finitely represented in $X$ but as far as I can see this is not good enough (or is it?).  
My question:  If a space is reflexive and does not contain $\ell_1$ finitely represented is it necessarily uniformly convex?  
I suspect the answer is no but I don't have a counterexample.  
Another question: Are there any known conditions on a basis, which (1) do not imply the basis is unconditional and (2) do imply the space is uniformly convex?

REPLY [2 votes]: I think James also showed that if $X$ does not contain almost isometric copies of $\ell_1^2$ (he called such a space uniformly non-square) then $X$ is superreflexive. This is no longer true for $n>2$, as James later constructed a non-reflexive, uniformly non-octahedral (no almost isometric copies of $\ell_1^3$) space, thus also having non-trivial type. 
Maybe you can check whether your space is uniformly non-square. Connecting it with your last question I think that you would have to verify that $\exists \delta>0$ such that for any normalized block vectors $x$ and $y$ (but not necessarily disjointly supported) there exist a choice of signs such that $||x\pm y||<2-\delta.$ I don't think this condition implies unconditionality. 
Hopefully this makes sense...