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.