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

Alonso-Gutiérrez, David; Prochno, Joscha

doi:10.1090/S0002-9939-2014-12401-4

On the Gaussian behavior of marginals and the mean width of random polytopes
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by David Alonso-Gutiérrez and Joscha Prochno

Proc. Amer. Math. Soc. 143 (2015), 821-832

DOI: https://doi.org/10.1090/S0002-9939-2014-12401-4

Published electronically: November 3, 2014

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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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