Computational geometry column 56
ACM SIGACT News
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According to a classical result of Grünbaum, the transversal number τ(ℱ) of any family ℱ of pairwise-intersecting translates or homothets of a convex body C in ℝd is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ(ℱ) to the packing number ν(ℱ) over all finite families ℱ of translates (resp. homothets) of a convex body C in ℝd . Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝd , and gave the first bounds on α(C) for convex bodies C in ℝd and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in ℝd , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝd 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.