Piercing Translates and Homothets of a Convex Body

According to a classical result of Gr¨unbaum, 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 ...

• Type: Conference
• Year: 2009
• Where: ESA
• Authors: Adrian Dumitrescu, Minghui Jiang
• Keywords: Algorithms, Convex Bodies, Convex Body, ESA 2009, Transversal Number