“High-dimensional random polytopes of the Beta kind”
Venerdì 31 Marzo 2023, ore 14:30 - Aula 1BC45 e Zoom - Nicola Turchi (Università 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.
A prominent reason for the interest surrounding them is that the beta distribution exhibits peculiar properties that make exact computations possible to achieve, contrarily to most other models of randomness with non-independent coordinates.
As the space dimension grows, the expected fraction of the volume that these polytopes fill within their supporting balls can be asymptotically negligible or not, depending on the number of points which are picked 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 in average. Lastly, we show how we can handle more precise asymptotics for the approach towards zero in the lower regimes of points, based on a work in progress with G. Bonnet and Z. Kabluchko.