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Discrete & Computational Geometry

, Volume 59, Issue 4, pp 990–1000

# Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary

• Christoph Thäle
Article

## Abstract

Given a convex body K with smooth boundary $$\partial K$$, select a fixed number n of uniformly distributed random points from $$\partial K$$. The convex hull $$K_n$$ of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of $$K_n$$ to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.

## Keywords

Central limit theorem Random polytope Surface body Stochastic geometry

## Mathematics Subject Classification

52A22 60D05 60F05

## Notes

### Acknowledgements

I am grateful to Julian Grote (Bochum) for stimulating discussions and helpful comments. I also thank two anonymous referees for careful reading and their suggestions.

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## Copyright information

© Springer Science+Business Media New York 2017

## Authors and Affiliations

1. 1.Faculty of MathematicsRuhr University BochumBochumGermany