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 $(\beta)$ of Rolewicz. In particular, we give a "coordinate-free" proof of the fact that every Banach space with property $(\beta)$ 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 $(\beta)$ is positive at some point in the interval $(0,1)$, then $X$ admits an equivalent norm with property $(\beta)$. 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.