When is the Szlenk derivation of a dual unit ball another ball?

TL;DR

This paper investigates conditions under which Szlenk derivations of dual unit balls in Banach spaces are actual balls, revealing specific space properties that influence this geometric behavior.

Contribution

It identifies space properties that determine when Szlenk derivations are balls, including new results for spaces like Baernstein, Tsirelson, and Schlumprecht.

Findings

All ε-Szlenk derivations are balls in spaces with property (M*)

In Baernstein's space, derivations are balls with the same radius as ℓ₂

Derivations are not balls in Tsirelson's and Schlumprecht's spaces

Abstract

We show that if a separable Banach space has Kalton's property $(M^\ast)$, then all $\varepsilon$-Szlenk derivations of the dual unit ball are balls, however, in the case of the dual of Baernstein's space, all those Szlenk derivations are balls having the same radius as for $\ell_2$, yet this space fails property $(M^\ast)$. By estimating the radii of enveloping balls, we show that the Szlenk derivations are not balls for Tsirelson's space and the dual of Schlumprecht's space. Using the Karush-Kuhn-Tucker theorem we prove that the same is true for the duals of certain sequential Orlicz spaces.