Threshold phenomena for high-dimensional random polytopes. (English) Zbl 1423.52007
It was shown by M. E. Dyer et al. [Random Struct. Algorithms 3, No. 1, 91–106 (1992; Zbl 0755.60013)] that the expected volume of the convex hull of \(N>n\) points chosen uniformly and independently from the vertices of the \(n\)-dimensional cube \([-1,1]^n\) exhibits a phase transition when \(N\) is taken to be exponential in the dimension \(n\) and when the space dimension \(n\) tends to infinity. This work inspired a number of subsequent papers.
The authors of the paper under review study \(N>n\) independent random points in \(\mathbb{R}^n\) distributed according to either the beta or beta-prime distribution. The main results of the paper establish threshold phenomena as the space dimension tends to infinity for measures on the convex hulls of \(N\) such points, including the volume and intrinsic volumes. In addition, the authors provide results for the dual setting of polytopes generated by random halfspaces.
The authors of the paper under review study \(N>n\) independent random points in \(\mathbb{R}^n\) distributed according to either the beta or beta-prime distribution. The main results of the paper establish threshold phenomena as the space dimension tends to infinity for measures on the convex hulls of \(N\) such points, including the volume and intrinsic volumes. In addition, the authors provide results for the dual setting of polytopes generated by random halfspaces.
Reviewer: Florian Pausinger (Belfast)
MSC:
| 52A23 | Asymptotic theory of convex bodies |
| 52B11 | \(n\)-dimensional polytopes |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 60D05 | Geometric probability and stochastic geometry |
Keywords:
beta distribution; beta-prime distribution; convex bodies; isotropic log-concave measures; phase transition; random polytopes; volume thresholdCitations:
Zbl 0755.60013References:
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