Covering the annulus of d-cube

Question

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c+x \mid c\in C \}$, for some $x\in R^d$, is called a homothetic copy of $C$. The number $\lambda > 0$ is called the coefficient of homothety.

Let $C$ be a $d$-dimensional cube of side length 1. Now draw another cube $C'$ concentric and homothetic with respect to $C$ and having $\lambda= 1/2-\epsilon$ (or equivalently, $C'$ is of side $1/2-\epsilon$).

Now, my questions are:

1. How many translated copies of $C'$ would be required to cover the annulus obtained between $C'$ and $C$?

2. How many translated copies of $C'$ would be required to cover $C$?

Answer 1

$3^d-1$ and $3^d$, resp. Let $C=[0,1]^d$, Consider the $3^d$ points in $C$ all whose coordinates are from the set $\{0,\frac12,1\}$. No translated copy of $C'$ can cover two of these points, hence at least $3^d$ copies of $C'$ are required. On the other hand, a covering by $3^d$ copies of $C'$ (with the requirement that one of them is concentric to $C$) is easy to construct by induction in $d$.