On various diametral notions of points in the unit ball of some vector-valued function spaces

In this article, we study the ccs-Daugavet, ccs-$\mathrm{\Delta }$, super-Daugavet, super-$\mathrm{\Delta }$, Daugavet, $\mathrm{\Delta }$, and $\mathrm{\nabla }$ points in the unit balls of vector-valued function spaces $C_0(L,X)$, $A(K,X)$, $L_{\mathrm{\infty }}(\mu ,X)$, and $L_1(\mu ,X)$. To partially or fully characterize these diametral points, we first provide improvements of several stability results under ${\oplus }_{\mathrm{\infty }}$ and ${\oplus }_1$-sums shown in the literature. For complex Banach spaces, $\mathrm{\nabla }$ points are identical to Daugavet points, and so the study of $\mathrm{\nabla }$ points only makes sense when a Banach space is real. Consequently, we obtain that the seven notions of diametral points are equivalent for $L_{\mathrm{\infty }}(\mu )$ and uniform algebra when $K$ is infinite.