Best and worst centrally symmetric convex covering shapes

Question

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.

Answer 1

Q2: there is no worst shape, but arbitrarily bad shapes. Take a very thin and long rectangle: then any intersection of a displacement of it with the blue square has area bounded by some $a$, which can be made arbitrarily small. In consequence, at least $1/a$ copies are needed in any case to cover the blue square.

Variant of the question: search $C$ maximizing $\inf_A \rho(A\cdot C)$ where $A$ runs over linear maps.