Corpus ID: 208513935

# Phase transition for the volume of high-dimensional random polytopes

@article{Bonnet2019PhaseTF,
title={Phase transition for the volume of high-dimensional random polytopes},
author={Gilles Bonnet and Zakhar Kabluchko and Nicola Turchi},
journal={arXiv: Probability},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Probability
• The beta polytope $P_{n,d}^\beta$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^{\beta}$ if $\beta>-1$ (in particular, $\beta=0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $\beta=-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous… CONTINUE READING

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