Expectation of intrinsic volumes of random polytopes

Károly Böröczky; Lars Hoffmann; Daniel Hug

doi:10.1007/s10998-008-8143-4

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 57: Issue 2

Expectation of intrinsic volumes of random polytopes

Authors:

Károly Böröczky

Károly Böröczky

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Lars Hoffmann

Lars Hoffmann Technische Universität Wien Institut für Diskrete Mathematik und Geometrie Wiedner Hauptstr. 8-10 A-1040 Wien Austria

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Daniel Hug

Daniel Hug Universität Karlsruhe (TH), KIT Institut für Algebra und Geometrie D-76133 Karlsruhe Germany

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Pages:: 143–164

Online Publication Date:: 10 Dec 2008

Publication Date:: 01 Dec 2008

Article Category:: Research Article

DOI:: https://doi.org/10.1007/s10998-008-8143-4

Keywords:: 52A20; 52A22; 60D05; random polytope; expectation; intrinsic volume; cap covering; convolution body; approximation

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Abstract

Let K be a convex body in ℝ^d, let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)