On the Smoothed Complexity of Convex Hulls

Authors Olivier Devillers, Marc Glisse, Xavier Goaoc, Rémy Thomasse

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Olivier Devillers
Marc Glisse
Xavier Goaoc
Rémy Thomasse

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Olivier Devillers, Marc Glisse, Xavier Goaoc, and Rémy Thomasse. On the Smoothed Complexity of Convex Hulls. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 224-239, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})). We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise.
  • Probabilistic analysis
  • Worst-case analysis
  • Gaussian noise


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