23 research outputs found

    The intersection graph of the disks with diameters the sides of a convex n-gon

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    © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Given a convex polygon of sides, one can draw disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the disks and two disks are adjacent if and only if they have a point in common. We introduce the study of this graph by proving that it is planar for every convex polygon.Postprint (author's final draft

    Proper Coloring of Geometric Hypergraphs

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    We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions

    Proper coloring of geometric hypergraphs

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    We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m = 3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions. © Balázs Keszegh and Dömötör Pálvölgyi

    Extending the Centerpoint Theorem to Multiple Points

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    The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median. We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)

    Proper Coloring of Geometric Hypergraphs

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    We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions

    Hyperbolic billiards with nearly flat focusing boundaries. I

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    The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstardard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms.Comment: 21 pages, 9 figure

    Master index of volumes 61–70

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