  This work revolves around the two following questions: Given a convex body
$C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set
$S\subset\mathbb{R}^d$ (or a finite Borel measure $\mu$ on $\mathbb{R}^d$), how
many homothets of $C$ are required to cover $S$ if no homothet is allowed to
cover more than $k$ points of $S$ (or have measure larger than $k$)? How many
homothets of $C$ can be packed if each of them must cover at least $k$ points
of $S$ (or have measure at least $k$)? We prove that, so long as $S$ is not too
degenerate, the answer to both questions is $\Theta_d(\frac{|S|}{k})$, where
the hidden constant is independent of $d$. This is optimal up to a
multiplicative constant. Analogous results hold in the case of measures. Then
we introduce a generalization of the standard covering and packing densities of
a convex body $C$ to Borel measure spaces in $\mathbb{R}^d$ and, using the
aforementioned bounds, we show that they are bounded from above and below,
respectively, by functions of $d$. As an intermediate result, we give a simple
proof the existence of weak $\epsilon$-nets of size $O(\frac{1}{\epsilon})$ for
the range space induced by all homothets of $C$. Following some recent work in
discrete geometry, we investigate the case $d=k=2$ in greater detail. We also
provide polynomial time algorithms for constructing a packing/covering
exhibiting the $\Theta_d(\frac{|S|}{k})$ bound mentioned above in the case that
$C$ is an Euclidean ball. Finally, it is shown that if $C$ is a square then it
is NP-hard to decide whether $S$ can be covered using $\frac{|S|}{4}$ squares
containing $4$ points each.
