Piercing translates and homothets of a convex body

According to a classical result of Gr"unbaum, the transversal number $\tau (\text{F})$ of any family $\text{F}$ of pairwise-intersecting translates or homothets of a convex body $C$ in ${\text{RR}}^d$ is bounded by a function of $d$. Denote by $\alpha (C)$ (resp. $\beta (C)$) the supremum of the ratio of the transversal number $\tau (\text{F})$ to the packing number $\nu (\text{F})$ over all families $\text{F}$ of translates (resp. homothets) of a convex body $C$ in ${\text{RR}}^d$. Kim et al. recently showed that $\alpha (C)$ is bounded by a function of $d$ for any convex body $C$ in ${\text{RR}}^d$, and gave the first bounds on $\alpha (C)$ for convex bodies $C$ in ${\text{RR}}^d$ and on $\beta (C)$ for convex bodies $C$ in the plane. Here we show that $\beta (C)$ is also bounded by a function of $d$ for any convex body $C$ in ${\text{RR}}^d$, and present new or improved bounds on both $\alpha (C)$ and $\beta (C)$ for various convex bodies $C$ in ${\text{RR}}^d$ for all dimensions $d$. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.