Asymptotic Properties of Random Polytopes
Random polytopes have constituted some of the central objects of stochastic geometry for more than 150 years. They are in general generated as convex hulls of a random set of points in the Euclidean space. The study of such models requires the use of ingredients coming from both convex geometry and probability theory. In the last decades, the study has been focused on their asymptotic properties and in particular expectation and variance estimates. In several joint works with Tomasz Schreiber and J. E. Yukich, we have investigated the scaling limit of several models (uniform model in the unit-ball, uniform model in a smooth convex body, Gaussian model) and have deduced from it limiting variances for several geometric characteristics including the number of k-dimensional faces and the volume. In this paper, we survey the most recent advances on these questions and we emphasize the particular cases of random polytopes in the unit-ball and Gaussian polytopes.
KeywordsConvex Hull Central Limit Theorem Convex Body Gaussian Model Variance Asymptotics
- 9.Blaschke, W.: Über affine Geometrie XI: Losung des Vierpunk- tproblemsvon Sylvester aus der Theorie der geometrischen Wahrschein- lichkeiten. Ber. Verh. S achs. Ges. Wiss. Leipzig, Math.-Phys. Kl. 69, 436–453 (1917)Google Scholar
- 12.Calka, P., Yukich, J.E.: Variance asymptotics and scaling limits for Gaussian Polytopes. Probability Theory and Related Fields (2015, to appear). http://arxiv.org/abs/1403.1010/
- 20.Reitzner, M.: Random polytopes. In: Kendall, W.S., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–77. Oxford University Press, Oxford (2010)Google Scholar