  We introduce slicely countably determined points (SCD points) of a bounded
and convex subset of a Banach space which extends the notions of denting
points, strongly regular points and much more. We completely characterize SCD
points in the unit balls of $L_1$-preduals. We study SCD points in direct sums
of Banach spaces and obtain that an infinite sum of Banach spaces may have an
SCD point despite the fact that none of its components have it. We then prove
sufficient conditions to get that an elementary tensor $x\otimes y$ is an SCD
point in the unit ball of the projective tensor product $X
\widehat{\otimes}_\pi Y$. Regarding Lipschitz-free spaces on compact metric
spaces, we show that norm one SCD points of their unit balls are exactly the
ones that can be approximated by convex combinations of strongly exposed points
of the unit ball. Finally, as applications, we prove a new inheritance result
for the Daugavet property to its subspaces, we show that separable Banach
spaces for which every convex series of slices intersects the unit sphere must
contain an isomorphic copy of $\ell_1$, and we get pointwise conditions on an
operator on a Banach space with the Daugavet property to satisfy the Daugavet
equation.
