  Inspired by R. Whitley's thickness index the last named author recently
introduced the Daugavet index of thickness of Banach spaces. We continue the
investigation of the behavior of this index and also consider two new versions
of the Daugavet index of thickness, which helps us solve an open problem which
connect the Daugavet indices with the Daugavet equation. Moreover, we will
improve the formerly known estimates of the behavior of Daugavet index on
direct sums of Banach spaces by establishing sharp bounds. As a consequence of
our results we prove that, for every $0<\delta<2$, there exists a Banach space
where the infimum of the diameter of convex combinations of slices of the unit
ball is exactly $\delta$, solving an open question from the literature.
Finally, we prove that an open question posed by Ivakhno in 2006 about the
relation between the radius and diameter of slices has a negative answer.
