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On the Gaussian behavior of marginals and the mean width of random polytopes


Authors: David Alonso-Gutiérrez and Joscha Prochno
Journal: Proc. Amer. Math. Soc. 143 (2015), 821-832
MSC (2010): Primary 52A22; Secondary 52A23, 05D40, 46B09
DOI: https://doi.org/10.1090/S0002-9939-2014-12401-4
Published electronically: November 3, 2014
MathSciNet review: 3283668
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Abstract: We show that the expected value of the mean width of a random polytope generated by $ N$ random vectors ( $ n\leq N\leq e^{\sqrt n}$) uniformly distributed in an isotropic convex body in $ \mathbb{R}^n$ is of the order $ \sqrt {\log N} L_K$. This completes a result of Dafnis, Giannopoulos and Tsolomitis. We also prove some results in connection with the 1-dimensional marginals of the uniform probability measure on an isotropic convex body, extending the interval in which the average of the distribution functions of those marginals behaves in a sub- or supergaussian way.


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Additional Information

David Alonso-Gutiérrez
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, 505 Central Academic Building, Edmonton T6G 2G1, Canada
Address at time of publication: Departament de Matemàtiques, Universitat Jaume I, Campus de Riu Sec, E-12071 Castelló de la Plana, Spain
Email: alonsod@uji.es

Joscha Prochno
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, 605 Central Academic Building, Edmonton T6G 2G1, Canada
Address at time of publication: Institute of Analysis, Johannes Keples University, Liuz Altunbergerstr. 69, 4040 Liuz, Austria
Email: prochno@ualberta.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12401-4
Keywords: Supergaussian direction, random polytope, Orlicz norm, mean width
Received by editor(s): May 28, 2012
Received by editor(s) in revised form: March 22, 2013
Published electronically: November 3, 2014
Communicated by: David Levin
Article copyright: © Copyright 2014 American Mathematical Society