The support function of the high-dimensional Poisson polytope

Let $K_{\lambda }^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $\lambda$ in ${\mathbb{R}}^d$, $d\ge 2$, with the Euclidean unit ball ${\mathbb{B}}^d$. In this paper, we study the asymptotic behavior as $d\to \mathrm{\infty }$ of the support function $h_{\lambda }^{(d)}(u):={\sup }_{x\in K_{\lambda }^d}⟨u,x⟩$ in an arbitrary direction $u\in {\mathbb{S}}^{d-1}$ of the Poisson polytope $K_{\lambda }^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $\lambda :=\lambda (d)$ and derive in each regime the precise distributional convergence of $h_{\lambda }^{(d)}$ 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.