A covering index for Banach spaces
Matias Raja
TL;DR
This work introduces the covering index Θ_X for real Banach spaces, defined via finite convex coverings of the unit ball and the essential inradius, and studies its isomorphic behavior and asymptotics. By relating Θ_X to the asymptotic moduli of uniform convexity (AUC) and uniform smoothness (AUS), the authors develop upper bounds using FDD decompositions and q-AUC renormings, and lower bounds via Szlenk-derived sets and AUS renormings. They show Θ_X(n) ≍ n^{-1/p} for spaces embeddable into ℓ_p-sums of finite-dimensional spaces and for spaces admitting both p-AUC and p-AUS renormings, with Θ_{ℓ_p}(n) ≍ n^{-1/p} and Θ_{T^*}(n) ≥ 1/2 in the Tsirelson framework. The paper highlights the potential of Θ_X to distinguish classical spaces, discusses parallels with entropy numbers, and poses several open problems including exact values and the precise role of moduli in Θ_X’s behavior.
Abstract
We introduce a new isomorphic quantity for Banach spaces, the index $Θ_X$, based on finite convex coverings of the unit ball. This index is closely related to the asymptotic moduli of uniform convexity and uniform smoothness, so that it can be calculated for several classical Banach spaces.