Moduli of convexity and smoothness of reflexive subspaces of L1

We show that for any probability measure μ there exists an equivalent norm on the space L1(μ) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces. © 2011 Elsevier Inc. All rights reserved.