Moduli of convexity and smoothness of reflexive subspaces of L^1

TL;DR

The paper demonstrates that for any probability measure, an equivalent norm on L^1 can be constructed so that all reflexive subspaces are uniformly smooth and convex with specific moduli, enhancing geometric understanding.

Contribution

It introduces a new renorming of L^1 spaces ensuring uniform smoothness and convexity in all reflexive subspaces, with explicit modulus estimates.

Findings

Existence of an equivalent norm on L^1() for any probability measure.

Reflexive subspaces are uniformly smooth and convex with power type 2 moduli.

Provides estimates for the modulus of smoothness of these subspaces.

Abstract

We show that for any probability measure \mu there exists an equivalent norm on the space L^1(\mu) 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.