Uniformly convex renormings and generalized cotypes

TL;DR

This paper investigates how renormings can improve the modulus of convexity in super-reflexive Banach spaces, establishing bounds related to generalized cotypes and showing conditions for optimal power type 2 convexity.

Contribution

It provides explicit bounds for the modulus of convexity under renormings and links these bounds to generalized cotypes and the structure of UMD spaces.

Findings

Explicit upper bounds for modulus of convexity under renormings

Spaces with bounds equivalent to t^2 admit power type 2 renormings

UMD spaces can have renormings with convexity modulus proportional to their cotype

Abstract

We are concerned about improvements of the modulus of convexity by renormings of a super-reflexive Banach space. Typically optimal results are beyond Pisier's power functions bounds $t^p$, with $p \geq 2$, and they are related to the notion of generalized cotype. We obtain an explicit upper bound for all the modulus of convexity of equivalent renormings and we show that if this bound is equivalent to $t^2$, the best possible, then the space admits a renorming with modulus of power type $2$. We show that a UMD space admits a renormings with modulus of convexity bigger, up to a multiplicative constant, than its cotype. We also prove the super-multiplicativity of the supremum of the set of cotypes.