An asymptotic analog of a local-to-global phenomenon for uniformly convex renormings

Abstract

In this note, we investigate the renorming theory of Banach spaces with property $(β)$ of Rolewicz. In particular, we give a "coordinate-free" proof of the fact that every Banach space with property $(β)$ admits an equivalent norm that is asymptotically uniformly smooth; a result originally due to Kutzarova for spaces with a Schauder basis. We also show that if a natural modulus associated with a Banach space $X$ with property $(β)$ is positive at some point in the interval $(0,1)$, then $X$ admits an equivalent norm with property $(β)$. This is an asymptotic analog of a profound result from the local geometry of Banach spaces that states that if the modulus of uniform convexity of a Banach space $X$ is positive at some point in the interval $(0,2)$, then $X$ admits an equivalent norm that is uniformly convex.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Corollary 2.2
• Theorem 2.3
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• proof