Define an open $k$-sector of a disk as the portion between two radii separated
by an angle of $2\pi/k$, but open along the two radii (and closed along the
circle boundary).
Call a set a *sub-$k$-sector* if it is a closed set and a subset of an open $k$-sector.
So an open $2$-sector is a half-disk minus the diameter,
and a sub-$2$-sector is a closed set that fits inside this set.

^{ Left: Coverage by three sub-$2$-sectors. Right: Four sub-$4$-sectors. Diameters are dotted lines. }

It is clear that one can position three sub-$2$-sectors so that they cover a disk (above, left).

. Is it correct that three sub-semiballs suffice to cover a ball in $\mathbb{R}^3$? What is the generalization to $\mathbb{R}^d$?Q1

By a sub-semiball I mean the natural analog: a closed subset of a half-ball minus its bounding plane.

. Does it require six sub-$4$-sectors to cover a disk?Q2

I would like to believe that $k{+}1$ sub-$k$-sectors suffice—-so little of the quadrant is missing!—but the drawing above suggests otherwise.

**Update**. Here is my attempt to illustrate Douglas Zare's answer to

**.**

*Q2*