Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw
down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$,
until $S$ is entirely covered, i.e., every point of $S$ lies in some copy of $C$.
For example, below (left) it took $9$ disks to cover the blue unit square, and (right)
$10$ squares of the same area to cover the blue unit square (scaled differently):

Let $\rho(C)$ be the expected number of such randomly placed copies of $C$ needed to
cover $S$. My questions are:

**Q1**. Is the disk the most efficient such covering shape, in that it minimizes $\rho(C)$ over all centrally symmetric convex bodies $C$ of the same area? If not, which shape is the best?**Q2**. What shape is the worst covering shape, achieving the maximum of $\rho(C)$ over such shapes?**Q3**. Do the best and worst shapes depend upon the choice to cover a square rather than to cover some other convex shape?

The same questions can be asked in any dimension.

fullycovering $S$, is it not equivalent to think about the expected amount of $S$ covered by randomly placing a single copy of $C$? $\endgroup$ – Benjamin Dickman Sep 27 '13 at 8:46