## Electronic Journal of Probability

### Approximation of smooth convex bodies by random polytopes

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 9, 21 pp.

Dates
Accepted: 22 December 2017
First available in Project Euclid: 12 February 2018

https://projecteuclid.org/euclid.ejp/1518426057

Digital Object Identifier
doi:10.1214/17-EJP131

Mathematical Reviews number (MathSciNet)
MR3771746

Zentralblatt MATH identifier
1395.52008

#### Citation

Grote, Julian; Werner, Elisabeth. Approximation of smooth convex bodies by random polytopes. Electron. J. Probab. 23 (2018), paper no. 9, 21 pp. doi:10.1214/17-EJP131. https://projecteuclid.org/euclid.ejp/1518426057

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