Renormings via asymptotic uniform convexity and the approximation property on near Hilbertian spaces

Let X be a Banach space and let (O, A , P) be the probability space with O = -1,1N ,A the Borel sigma-algebra, and P the Haar measure on O. If the space of X-valued strongly-measurable Bochner-integrable functions Lr(O, X), 1 < r < infinity, has an equivalent asymptotically uniformly convex norm of power type p then X admits a uniformly convex renorming of power type.; Using A. M. Davie's construction of a subspace of ℓp , p > 2, without the approximation property, it is shown that there exists an asymptotically Hilbertian space E which is 'almost' a weak Hilbert space and that fails the approximation property. In addition, the space E is a subspace of a space with an unconditional basis and can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.