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Transactions of the American Mathematical Society

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Asymptotic normality for random polytopes in non-Euclidean geometries


Authors: Florian Besau and Christoph Thäle
Journal: Trans. Amer. Math. Soc. 373 (2020), 8911-8941
MSC (2010): Primary 52A22, 52A55; Secondary 60D05, 60F05
DOI: https://doi.org/10.1090/tran/8217
Published electronically: October 5, 2020
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Abstract: Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak's dual Brunn-Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein's method for normal approximation and geometric properties of weighted floating bodies.


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Additional Information

Florian Besau
Affiliation: Department of Mathematics, Vienna University of Technology, Vienna, Austria
MR Author ID: 1174501
Email: florian.besau@tuwien.ac.at

Christoph Thäle
Affiliation: Department of Mathematics, Ruhr University Bochum, Bochum, Germany
Email: christoph.thaele@rub.de

DOI: https://doi.org/10.1090/tran/8217
Keywords: Central limit theorem, dual Brunn--Minkowski theory, dual volume, floating body, Hilbert geometry, hyperbolic space, random polytope, spherical space, Stein's method, stochastic geometry, weighted floating body
Received by editor(s): September 12, 2019
Received by editor(s) in revised form: May 24, 2020
Published electronically: October 5, 2020
Article copyright: © Copyright 2020 American Mathematical Society