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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.00507 |
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| _version_ | 1866914961439064064 |
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| author | Calka, Pierre Dadoun, Benjamin |
| author_facet | Calka, Pierre Dadoun, Benjamin |
| contents | Let $K_λ^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $λ$ in $\mathbb{R}^d$, $d \ge 2$, with the Euclidean unit ball $\mathbb{B}^d$. In this paper, we study the asymptotic behavior as $d\to\infty$ of the support function $h_λ^{(d)}(u) :=\sup_{x\in K_λ^d}\langle u,x\rangle$ in an arbitrary direction $u \in {\mathbb S}^{d-1}$ of the Poisson polytope $K_λ^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $λ:=λ(d)$ and derive in each regime the precise distributional convergence of $h_λ^{(d)}$ after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once. We finally deduce partial counterparts for the radius-vector function of the polytope. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The support function of the high-dimensional Poisson polytope Calka, Pierre Dadoun, Benjamin Probability Let $K_λ^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $λ$ in $\mathbb{R}^d$, $d \ge 2$, with the Euclidean unit ball $\mathbb{B}^d$. In this paper, we study the asymptotic behavior as $d\to\infty$ of the support function $h_λ^{(d)}(u) :=\sup_{x\in K_λ^d}\langle u,x\rangle$ in an arbitrary direction $u \in {\mathbb S}^{d-1}$ of the Poisson polytope $K_λ^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $λ:=λ(d)$ and derive in each regime the precise distributional convergence of $h_λ^{(d)}$ after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once. We finally deduce partial counterparts for the radius-vector function of the polytope. |
| title | The support function of the high-dimensional Poisson polytope |
| topic | Probability |
| url | https://arxiv.org/abs/2410.00507 |