On a Problem of Danzer

Abstract

LetCbe a bounded convex object in ℝd, and letPbe a set ofnpoints lying outsideC. Further, letcp,cqbe two integers with 1 ⩽cq⩽cp⩽n- ⌊d/2⌋, such that everycp+ ⌊d/2⌋ points ofPcontain a subset of sizecq+ ⌊d/2⌋ whose convex hull is disjoint fromC. Then our main theorem states the existence of a partition ofPinto a small number of subsets, each of whose convex hulls are disjoint fromC. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p,q) numbers for balls in ℝd. For example, it follows from our theorem that whenp>q= (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p,q)-property can be hit byO((1+β)2d2p1+1/βlogp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughlyO(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p,q) for convex sets in ℝdfor various ranges ofpandq, a polynomial bound is obtained for regions with low union complexity in the plane.