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?