Piercing Translates and Homothets of a Convex Body

According to a classical result of Grunbaum, the transversal number tau(F) of any family F of pairwise-intersecting translates or homothets of a convex body C in a"e (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(F) to the packing number nu(F) over all finite families F of translates (resp. homothets) of a convex body C in a"e (d) . Kim et al. recently showed that alpha(C) is bounded by a function of d for any convex body C in a"e (d) , and gave the first bounds on alpha(C) for convex bodies C in a"e (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 a"e (d) , and present new or improved bounds on both alpha(C) and beta(C) for various convex bodies C in a"e (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.