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Let P be a set of n points in R d. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than dn d+1 points of P. We call a point x a strong centerpoint for a family of objects C if x ∈ P is contained in every object C ∈ C that contains more than a constant fraction of points of P. A strong centerpoint does… (More)

We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type… (More)

We say a family of geometric objects C has (l, k)-property if every subfamily C ⊆ C of cardinality at most l is k-piercable. In this paper we investigate the existence of g(k, d) such that if any family of objects C in R d has the (g(k, d), k)-property, then C is k-piercable. Danzer and Grünbaum showed that g(k, d) is infinite for families of boxes and… (More)

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In… (More)

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers $$k_\ell $$ k ℓ and $$k_r$$ k r , and a family $$\mathcal F $$ F of $$\ell $$… (More)

Let P be a set of n points in R d and F be a family of geometric objects. We call a point x ∈ P a strong centerpoint of P w.r.t F if x is contained in all F ∈ F that contains more than cn points from P , where c is a fixed constant. A strong centerpoint does not exist even when F is the family of halfspaces in the plane. We prove the existence of strong… (More)

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers k and k r , and a family F of sets over U , the Gen-RBSC problem is to decide… (More)