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- W1562781560 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." @default.
- W1562781560 created "2016-06-24" @default.
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- W1562781560 date "1997-07-01" @default.
- W1562781560 modified "2023-09-27" @default.
- W1562781560 title "Weak ɛ-Nets for Points on a Hypersphere" @default.
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- W1562781560 doi "https://doi.org/10.1007/pl00009309" @default.
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