Planar realizations of nonlinear Davenport-Schinzel sequences by segments
Discrete & Computational Geometry
Disjoint simplices and geometric hypergraphs
Proceedings of the third international conference on Combinatorial mathematics
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
The colored Tverberg's problem and complexes of injective functions
Journal of Combinatorial Theory Series A
Ray shooting and parametric search
SIAM Journal on Computing
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Colourful linear programming and its relatives
Mathematics of Operations Research
Ray shooting and lines in space
Handbook of discrete and computational geometry
Lectures on Discrete Geometry
On the Boundary Complexity of the Union of Fat Triangles
SIAM Journal on Computing
Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
The colourful feasibility problem
Discrete Applied Mathematics
Combinatorica
On Approximate Range Counting and Depth
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Discrete & Computational Geometry - Special Issue Dedicated to the Memory of Victor Klee
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d@?N, the maximum combinatorial complexity of the colorful union of n colored points in R^d is between @W(n^(^d^-^1^)^^^2) and O(n^(^d^-^1^)^^^2logn). For d=2, the upper bound is known to be O(n), and for d=3 we present an upper bound of O(n^4@a(n)), where @a(@?) is the extremely slowly growing inverse Ackermann function. We also prove several structural properties of the colorful union. In particular, we show that the boundary of the colorful union is covered by O(n^d^-^1) hyperplanes, and the colorful union is the union of d+1 star-shaped polyhedra. These properties lead to efficient data structures for point inclusion queries in the colorful union.