Daugavet property in projective symmetric tensor products of Banach spaces

Abstract We show that all the symmetric projective tensor products of a Banach space X have Daugavet property provided has and either is an $$L_1$$ L 1 -predual (i.e., $$X^{*}$$ X ? isometric to -space) or vector-valued -space. In process proving it, we get number results independent interest. For instance, characterise “localised” versions [i.e., points $$\Delta$$ ? -points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for -preduals terms extreme topological dual, result which allows polyhedrality real absence also provide new examples having convex diametral local diameter two property. These are applied nicely embedded spaces [in sense Werner (J Funct Anal 143:117–128, 1997)] so, particular, function algebras. Next, polynomial equivalent Lipschitz functions. Finally, improvement recent Rueda Zoca Inst Jussieu 20(4):1409–1428, 2021) about obtained.