Besau, F., Rosen, D., & Thäle, C. (2021). Random inscribed polytopes in projective geometries. Mathematische Annalen, 381(3–4), 1345–1372. https://doi.org/10.1007/s00208-021-02257-9
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie E104 - Institut für Diskrete Mathematik und Geometrie
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Journal:
Mathematische Annalen
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ISSN:
0025-5831
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Date (published):
Dec-2021
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Number of Pages:
28
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Publisher:
Springer
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Peer reviewed:
Yes
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Keywords:
central limit theorem; Hilbert geometry; random convex body; stochastic approximation; volume of a convex body
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Abstract:
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and a Berry–Esseen bound for functionals of independent random variables.
CC BY 4.0
