  A $\Delta$-point $x$ of a Banach space is a norm one element that is
arbitrarily close to convex combinations of elements in the unit ball that are
almost at distance $2$ from $x$. If, in addition, every point in the unit ball
is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A
Banach space $X$ has the Daugavet property if and only if every norm one
element is a Daugavet-point.
  We show that $\Delta$- and Daugavet-points are the same in $L_1$-spaces,
$L_1$-preduals, as well as in a big class of M\"untz spaces. We also provide an
example of a Banach space where all points on the unit sphere are
$\Delta$-points, but where none of them are Daugavet-points.
  We also study the property that the unit ball is the closed convex hull of
its $\Delta$-points. This gives rise to a new diameter two property that we
call the convex diametral diameter two property. We show that all $C(K)$
spaces, $K$ infinite compact Hausdorff, as well as all M\"untz spaces have this
property. Moreover, we show that this property is stable under absolute sums.
