**Preprint No.**
A-08-08

#
Vladimir Kadets, Varvara Shepelska,
Dirk Werner

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

**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 $\ell_1$.
**Keywords: **
Daugavet property, $\ell_1$-subspace, types, thickness

**Mathematics Subject Classification (MSC2000):** 46B04; 46B03, 46B25

**Language:** ENG
**Available:** Pr-A-08-08.ps,
Pr-A-08-08.ps.gz

**Contact**: Dirk Werner, Freie Universität Berlin, Fachbereich Mathematik und Informatik,

Arnimallee 6, D-14195 Berlin, Germany
(werner@math.fu-berlin.de)

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