Thickness of the unit sphere, $\ell_1$-types, and the almost Daugavet property

TL;DR

This paper characterizes Banach spaces where the unit sphere cannot be covered by finitely many small-radius balls, linking this property to $ ext{ell}_1$-type sequences and the almost Daugavet property, with a focus on spaces containing $ ext{ell}_1$.

Contribution

It provides a characterization of Banach spaces with a non-approximable unit sphere in terms of $ ext{ell}_1$-sequences and the almost Daugavet property, especially identifying those containing $ ext{ell}_1$.

Findings

Spaces with no finite $ ext{eps}$-net in the unit sphere are characterized.

Such spaces are exactly those containing an isomorphic copy of $ ext{ell}_1$.

The main result links geometric properties to the presence of $ ext{ell}_1$.

Abstract

We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separable Banach space $X$ is isomorphic to a space from this class if and only if $X$ contains an isomorphic copy of $\ell_1$.