Lower bounds for k-distance approximation

Quentin Mérigot (1)

1 MGMI - Modélisation Géométrique & Multirésolution pour l'Image

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.

Mots clés: Chernoff bound; order-k Voronoi diagram; k-distance

Domaines: Géométrie algorithmique [cs.CG]

Quentin Mérigot. Lower bounds for k-distance approximation. SoCG '13 - 29th ACM Symposium on Computational Geometry, Jun 2013, Rio de Janeiro, Brazil. pp.435-440, ⟨10.1145/2462356.2462367⟩. ⟨hal-00803410⟩