Abstract.
A weak ε -net for a set of points M, is a set of points W (not necessarily in M) where every convex set containing ε |M| points in M must contain at least one point in W. Weak ε-nets have applications in diverse areas such as computational geometry, learning theory, optimization, and statistics. Here we show that if M is a set of points quasi-uniformly distributed on a unit sphere S d-1, then there is a weak ε-net \(W \subseteq {\Bbb R}^d\) of size \(O(\log ({1}/{\epsilon}) \log ({1}/{\epsilon}))\) for M, where k d is exponential in d. A set of points M is quasi-uniformly distributed on S d-1 if, for any spherical cap \({\mbox{$\cal C$}} \subseteq S^{d-1}$ with $\mathop{\rm Vol}\nolimits({\mbox{$\cal C$}}) \geq c_1/|M|\) , we have \( c_2 \mathop{\rm Vol}\nolimits({\mbox{$\cal C$}}) \leq | {\mbox{$\cal C$}} \cap M| \leq c_3 \mathop{\rm Vol}\nolimits({\mbox{$\cal C$}}) \) for three positive constants c_1, c_2, and c 3 .
Further, we show that reducing our upper bound by asymptotically more than a \(\log(1/{\mbox{$\epsilon$}})\) factor directly implies the solution of a long unsolved problem of Danzer and Rogers.
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Received April 12, 1995, and in revised form May 8, 1995.
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Bradford, P., Capoyleas, V. Weak ɛ-Nets for Points on a Hypersphere. Discrete Comput Geom 18, 83–91 (1997). https://doi.org/10.1007/PL00009309
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DOI: https://doi.org/10.1007/PL00009309