Thickness of the unit sphere, ell₁-types, and the almost Daugavet property

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 $\ell_1$.