Weak E-nets for Points on a Hypersphere
نویسنده
چکیده
A weak e-net for a set of points M, is a set of points W (not necessarily in M) where every convex set containing eiMI points in M must contain at least one point in W. Weak e-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 sd-l, then there is a weak €-net w ~ JRd of size o((d 2 lo,d)k") log(kd/e)log(l/e)) for M, where kd is exponential in d. A set of points M is quasi-uniformly distributed on sd-l, if for any spherical cap C ~ sd-l with Vol(C) ~ c1/IMI, we have c2 Vol(C) ~ IC n Ml ~ c3 Vol(C) for three positive constants c1, c2, and c3. Further, we show that reducing our upper bound by asymptotically more than a log(l/e) factor directly implies the solution of a long unsolved problem of Danzer and Rogers.
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