  We show that all the symmetric projective tensor products of a Banach space
$X$ have the Daugavet property provided $X$ has the Daugavet property and
either $X$ is an $L_1$-predual (i.e.\ $X^*$ is isometric to an $L_1$-space) or
$X$ is a vector-valued $L_1$-space. In the process of proving it, we get a
number of results of independent interest. For instance, we characterise
"localised" versions of the Daugavet property (i.e.\ Daugavet points and
$\Delta$-points recently introduced) for $L_1$-preduals in terms of the extreme
points of the topological dual, a result which allows to characterise a
polyhedrality property of real $L_1$-preduals in terms of the absence of
$\Delta$-points and also to provide new examples of $L_1$-preduals having the
convex diametral local diameter two property. These results are also applied to
nicely embedded Banach spaces so, in particular, to function algebras. Next, we
show that the Daugavet property and the polynomial Daugavet property are
equivalent for $L_1$-preduals and for spaces of Lipschitz functions. Finally,
an improvement of recent results about the Daugavet property for projective
tensor products is also got.
