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On the Coarseness of Bicolored Point Sets
, 2012
"... Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = X ∩ R  − X ∩ B  be the bichromatic discrepancy of X. We say that a partitio ..."
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Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = X ∩ R  − X ∩ B  be the bichromatic discrepancy of X. We say that a partition Π = {S1, S2,...,Sk} of S is convex if the convex hulls of its members are pairwise disjoint. The discrepancy of a convex partition Π of S is the minimum ∇(Si) over the elements of Π. The coarseness of S is the discrepancy of the convex partition of S with maximum discrepancy. We study the coarseness of bicolored point sets, and relate it to well blended point sets. In particular, we show combinatorial results on the coarseness of general configurations and give efficient algorithms for computing the coarseness of two specific cases, namely when the set of points is in convex position and when the measure is restricted to convex partitions with two elements. 1
On the discrepancy for Cartesian products
"... Let B 2 denote the family of all circular discs in the plane. We prove that the discrepancy for the family fB 1 \Theta B 2 : B 1 ; B 2 2 B 2 g in R 4 is O(n 1=4+" ) for an arbitrarily small constant " ? 0, i.e. essentially the same as that for the family B 2 itself. The result is esta ..."
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Let B 2 denote the family of all circular discs in the plane. We prove that the discrepancy for the family fB 1 \Theta B 2 : B 1 ; B 2 2 B 2 g in R 4 is O(n 1=4+" ) for an arbitrarily small constant " ? 0, i.e. essentially the same as that for the family B 2 itself. The result is established for the combinatorial discrepancy, and consequently it holds for the discrepancy with respect to the Lebesgue measure as well. This answers a question of Beck and Chen. More generally, we prove an upper bound for the discrepancy for a family f Q k i=1 A i : A i 2 A i ; i = 1; 2; : : : ; kg, where each A i is a family in R d i , each of whose sets is described by a bounded number polynomial inequalities of bounded degree. The resulting discrepancy bound is determined by the "worst" of the families A i , and it depends on the existence of certain decompositions into constantcomplexity cells for arrangements of surfaces bounding the sets of A i . The proof uses Beck's partial coloring method ...
On stars and Steiner stars
, 2009
"... A Steiner star for a set P of n points in Rd connects an arbitrary point in Rd to all points of P, while a star connects one of the points in P to the remaining n − 1 points of P. All connections are realized by straight line segments. Let the length of a graph be the total Euclidean length of its e ..."
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A Steiner star for a set P of n points in Rd connects an arbitrary point in Rd to all points of P, while a star connects one of the points in P to the remaining n − 1 points of P. All connections are realized by straight line segments. Let the length of a graph be the total Euclidean length of its edges. Fekete and Meijer showed that the minimum star is at most √ 2 times longer than the minimum Steiner star for any finite point configuration in Rd. The supremum of the ratio between the two lengths, over all finite point configurations in Rd, is called the star Steiner ratio in Rd. It is conjectured that this ratio is 4/π = 1.2732... in the plane and 4/3 = 1.3333... in three dimensions. Here we give upper bounds of 1.3631 in the plane, and 1.3833 in 3space. These estimates yield improved upper bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. We also verify that the conjectured bound 4/π in the plane holds in two special cases. Our method exploits the connection with the classical problem of estimating the maximum sum of pairwise distances among n points on the unit sphere, first studied by László Fejes Tóth. It is quite general and yields the first nontrivial estimates below √ 2 on the star Steiner ratios in arbitrary dimensions. We show, however, that the star Steiner ratio in Rd tends to √ 2 as d goes to infinity. As it turns out, our estimates
Combinatorial discrepancy for boxes via the ellipsoidinfinity norm
"... The ellipsoidinfinity norm of a real m × n matrix A, denoted by ‖A‖E∞, is the minimum ` ∞ norm of an 0centered ellipsoid E ⊆ Rm that contains all column vectors of A. This quantity, introduced by the second author and Talwar in 2013, is polynomialtime computable and approximates the hereditary d ..."
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The ellipsoidinfinity norm of a real m × n matrix A, denoted by ‖A‖E∞, is the minimum ` ∞ norm of an 0centered ellipsoid E ⊆ Rm that contains all column vectors of A. This quantity, introduced by the second author and Talwar in 2013, is polynomialtime computable and approximates the hereditary discrepancy herdiscA as follows: ‖A‖E∞/O(logm) ≤ herdiscA ≤ ‖A‖E ∞ ·O( logm). Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality. We establish a number of favorable properties of ‖.‖E∞, such as the triangle inequality and multiplicativity w.r.t. the Kronecker (or tensor) product. We then demonstrate on several examples the power of the ellipsoidinfinity norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of Ω(logd−1 n) for the ddimensional Tusnády problem, asking for the combinatorial discrepancy of an npoint set in Rd with respect to axisparallel boxes. For d> 2, this improves the previous best lower bound, which was of order approximately log(d−1)/2 n, and it comes close to the best known upper bound of O(logd+1/2 n), for which we also obtain a new, very simple proof. 1