Weighted alpha shapes
Geometric Inference for Probability Measures
Foundations of Computational Mathematics
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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Consider a set P of N random points on the unit sphere of dimension d-1, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that after appropriate rescaling this halving polyhedron is Hausdorff close to the unit ball with high probability, as soon as the number of points grows like Omega(d log(d)). From this result, we deduce probabilistic lower bounds on the complexity of approximations of the distance to the empirical measure on the point set by distance-like functions.