Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
Improved bounds on weak &egr;-nets for convex sets
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Lectures on Discrete Geometry
Weak ε-nets have basis of size O (1/εlog(1/ε)) in any dimension
Computational Geometry: Theory and Applications
On Center Regions and Balls Containing Many Points
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
| Hi-index | 0.00 |
The colorful Carathéodory theorem [B82] states that given d+1 sets of points in Rd, the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the colorful Carathéodory theorem: given D2+1 sets of points in Rd, and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a D2-dimensional rainbow simplex intersecting C.