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
New Constructions of Weak ε-Nets
Discrete & Computational Geometry
Centerpoints and Tverberg's technique
Computational Geometry: Theory and Applications
A theorem of bárány revisited and extended
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Given a set P of n points in R^d and @e0, we consider the problem of constructing weak @e-nets for P. We show the following: pick a random sample Q of size O(1/@elog(1/@e)) from P. Then, with constant probability, a weak @e-net of P can be constructed from only the points of Q. This shows that weak @e-nets in R^d can be computed from a subset of P of size O(1/@elog(1/@e)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/@e. However, our final weak @e-nets still have a large size (with the dimension appearing in the exponent of 1/@e).