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2018 Approximation of smooth convex bodies by random polytopes
Julian Grote, Elisabeth Werner
Electron. J. Probab. 23: 1-21 (2018). DOI: 10.1214/17-EJP131

Abstract

Let $K$ be a convex body in $\mathbb{R} ^n$ and $f : \partial K \rightarrow \mathbb{R} _+$ a continuous, strictly positive function with $\int \limits _{\partial K} f(x) \mathrm{d} \mu _{\partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $\mathbb{R} ^n$ 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.

Citation

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Julian Grote. Elisabeth Werner. "Approximation of smooth convex bodies by random polytopes." Electron. J. Probab. 23 1 - 21, 2018. https://doi.org/10.1214/17-EJP131

Information

Received: 20 June 2017; Accepted: 22 December 2017; Published: 2018
First available in Project Euclid: 12 February 2018

zbMATH: 1395.52008
MathSciNet: MR3771746
Digital Object Identifier: 10.1214/17-EJP131

Subjects:
Primary: 52A22 , 60D05

Keywords: approximation , Convex bodies , Random polytopes

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