Phase transition for the volume of high-dimensional random polytopes

Turchi Nicola, Università degli Studi di Milano-Bicocca

Beta polytopes are a class of random polytopes, which arise as convex hulls of independent random points distributed according to a certain radially-symmetric probability distribution supported on the Euclidean ball, called the beta distribution. Examples are the uniform distribution in the ball and the uniform distribution on the sphere. As the space dimension grows, the expected fraction of the volume that these polytopes fill within their ambient balls can be asymptotically negligible or not, depending on the number of points which are chosen in each dimension. In this talk we give an overview on how to quantify this statement, first showing a rough threshold for the aforementioned growth and secondly a more precise one, namely how many points are needed to get any fraction of the volume in average. Lastly, we show how we can handle more precise asymptotics in the lower regimes of points. Other intrinsic volumes are also discussed. Based on works with G. Bonnet and Z. Kabluchko.

Area: CS41 - Limit theorems for random structures (Nicola Turchi)

Keywords: Beta distribution, convex hull, expected volume, phase transition, ran-dom polytopes

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