@article{ac6bd649836f48f3a3ace69bb24ead12,
title = "Phase transition for the volume of high-dimensional random polytopes",
abstract = "The beta polytope (Formula presented.) is the convex hull of n i.i.d. random points distributed in the unit ball of (Formula presented.) according to a density proportional to (Formula presented.) if (Formula presented.) (in particular, (Formula presented.) corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if (Formula presented.). We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when (Formula presented.), their number of vertices.",
keywords = "Beta distribution, convex hull, expected volume, phase transition, random polytopes",
author = "Gilles Bonnet and Zakhar Kabluchko and Nicola Turchi",
note = "Funding Information: German Research Foundation under Germany's Excellence Strategy, Grant/Award Number: EXC 2044‐39068558; FNR grant FoRGES,R‐AGR‐3376‐10 Funding information Funding Information: We would like to thank Christoph Th{\"a}le (Bochum) for initiating this collaboration. The work of Zakhar Kabluchko has been supported by the German Research Foundation under Germany's Excellence Strategy EXC 2044‐390685587, Mathematics M{\"u}nster: Dynamics‐Geometry‐Structure. Nicola Turchi is supported by the FNR grant FoRGES (R‐AGR‐3376‐10) at Luxembourg University. Funding Information: information German Research Foundation under Germany's Excellence Strategy, Grant/Award Number: EXC 2044-39068558; FNR grant FoRGES,R-AGR-3376-10We would like to thank Christoph Th?le (Bochum) for initiating this collaboration. The work of Zakhar Kabluchko has been supported by the German Research Foundation under Germany's Excellence Strategy EXC 2044-390685587, Mathematics M?nster: Dynamics-Geometry-Structure. Nicola Turchi is supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University. Publisher Copyright: {\textcopyright} 2020 Wiley Periodicals LLC.",
year = "2021",
month = jul,
doi = "10.1002/rsa.20986",
language = "English",
volume = "58",
pages = "648--663",
journal = "Random structures & algorithms",
issn = "1042-9832",
publisher = "Wiley",
number = "4",
}