Weak ε-nets for points on a hypersphere

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 Sd-1, then there is a weak ε-net W ⊆ ℝd of size O(log(1/ε) log (1/ε)) for M, where kd is exponential in d. A set of points M is quasiuniformly distributed on Sd-1 if, for any spherical cap C ⊆ Sd-1 with Vol(C) ≥ c1/|M|, we have C2 Vol(C) ≤ |C ∩ M| ≤ 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(1/ε) factor directly implies the solution of a long unsolved problem of Danzer and Rogers.