4,028 research outputs found

    Small Strong Epsilon Nets

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    Let P be a set of n points in Rd\mathbb{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 dnd+1dn\over d+1 points of P. We call a point x a strong centerpoint for a family of objects C\mathcal{C} if xPx \in P is contained in every object CCC \in \mathcal{C} that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in R2\mathbb{R}^2. We prove that a strong centerpoint exists for axis-parallel boxes in Rd\mathbb{R}^d and give exact bounds. We then extend this to small strong ϵ\epsilon-nets in the plane and prove upper and lower bounds for ϵiS\epsilon_i^\mathcal{S} where S\mathcal{S} is the family of axis-parallel rectangles, halfspaces and disks. Here ϵiS\epsilon_i^\mathcal{S} represents the smallest real number in [0,1][0,1] such that there exists an ϵiS\epsilon_i^\mathcal{S}-net of size i with respect to S\mathcal{S}.Comment: 19 pages, 12 figure

    On Strong Centerpoints

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    Let PP be a set of nn points in Rd\mathbb{R}^d and F\mathcal{F} be a family of geometric objects. We call a point xPx \in P a strong centerpoint of PP w.r.t F\mathcal{F} if xx is contained in all FFF \in \mathcal{F} that contains more than cncn points from PP, where cc is a fixed constant. A strong centerpoint does not exist even when F\mathcal{F} is the family of halfspaces in the plane. We prove the existence of strong centerpoints with exact constants for convex polytopes defined by a fixed set of orientations. We also prove the existence of strong centerpoints for abstract set systems with bounded intersection
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