Stochastic sampling in computer graphics
ACM Transactions on Graphics (TOG)
On the intersection of edges of a geometric graph by straight lines
Discrete Mathematics
Halfspace range search: an algorithmic application of k-sets
Discrete & Computational Geometry
Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
On k-hulls and related problems
SIAM Journal on Computing
The maximum size of a convex polygon in a restricted set of points in the plane
Discrete & Computational Geometry
Coordinate representation of order types requires exponential storage
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
On the number of halving planes
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Points and triangles in the plane and halving planes in space
Discrete & Computational Geometry
An upper bound on the number of planar K-sets
Discrete & Computational Geometry
Convex independent sets and 7-holes in restricted planar point sets
Discrete & Computational Geometry
The colored Tverberg's problem and complexes of injective functions
Journal of Combinatorial Theory Series A
Counting triangle crossings and halving planes
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
The number of 3-sets of a finite point set in the plane
The number of 3-sets of a finite point set in the plane
On levels in arrangements of lines, segments, planes, and triangles
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Geometric matching under noise: combinatorial bounds and algorithms
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
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A halving hyperplane of a set S of n points in Rd contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most &dgr;n1/d, &dgr; some constant. Such a set S is called dense.In d=2 dimensions, the number of halving lines for a dense set can be as much as &OHgr;(nlogn), and it cannot exceed O(n5/4/log*n). The upper bound improves over the current best bound of O(n3/2/log*n) which holds more generally without any density assumption. In d=3 dimensions we show that O(n7/3) is an upper bound on the number of halving planes for a dense set. The proof is based on a metric argument that can be extended to d≥4 dimensions, where it leads to O(nd−2/d) as an upper bound for the number of halving hyperplanes.