Daugavet property in projective symmetric tensor products of Banach spaces

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 introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475-496 2020)] 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 [in the sense of Werner (J Funct Anal 143:117-128, 1997)] 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 in Rueda Zoca (J Inst Math Jussieu 20(4):1409-1428, 2021) about the Daugavet property for projective tensor products is also obtained.