Approximation of smooth convex bodies by random polytopes

2017/06/23 by Julian Grote, Grote, Julian, Elisabeth M. Werner +1 · 1 citation
Mathematics · #52A22 #60D05 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR) #math.MG #math.PR #msc:52A22 #msc:60D05

paper · pdf · doi:10.48550/arxiv.1706.07623

arxiv created 2017/07/06 · arxiv updated 2017/07/07

Abstract

Let K be a convex body in ℝⁿ and f : ∂ K → ℝ₊ a continuous, strictly positive function with ∫_{∂ K} f(x) d μ_{∂ K}(x) = 1. We give an upper bound for the approximation of K in the symmetric difference metric by an arbitrarily positioned polytope P_f in ℝⁿ having a fixed number of vertices. This generalizes a result by Ludwig, Schütt and Werner [36]. The polytope P_f is obtained by a random construction via a probability measure with density f. In our result, the dependence on the number of vertices is optimal. With the optimal density f, the dependence on K in our result is also optimal.

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