Variance asymptotics and scaling limits for random polytopes (Q329472)

Scaling limits for statistics of random polytopes is a long-studied problem in stochastic geometry with a rich history. A natural recipe to generate a random polytope is to throw points at random in a box, and then take their convex hull. Some statistics of interest include the number of \(k\)-faces \(f_k\) and the volume \(\mathrm{Vol}\) of this random polytope. The scaling limit problem is to establish precise convergence theorems for these statistics as the intensity of the points thrown go to infinity.NEWLINENEWLINEA more formal description is as follows: let \(P_\lambda\) be a homogeneous Poisson point process on \(\mathbb R^d\) with rate \(\lambda\). Fix an observation window \(K \subset \mathbb R^d\), take the convex hull of the points of \(P_\lambda\) that fall in \(K\). Call this random polytope \(K_\lambda\). It is generally believed that \(f_k(K_\lambda)\) and \(\mathrm{Vol}(K_\lambda)\) satisfy central limit theorems as \(\lambda \to \infty\) for most \(K\). This is first established for \(K\) with smooth boundary by \textit{M. Reitzner} [Probab. Theory Relat. Fields 133, No. 4, 483--507 (2005; Zbl 1081.60008)], and for \(K\) a convex polytope by \textit{I. Bárány} and \textit{M. Reitzner} [Adv. Math. 225, No. 4, 1986--2001 (2010; Zbl 1204.52007); Ann. Probab. 38, No. 4, 1507--1531 (2010; Zbl 1204.60018)]. However, for the latter case, the authors could not obtain precise asymptotics. This paper gives the explicit asymptotics when \(K\) is a simple polytope. The proof technique builds on the scaling transform \(T^{(\lambda)}\), put forward by \textit{Yu. Baryshnikov} [Probab. Theory Relat. Fields 117, No. 2, 163--182 (2000; Zbl 0961.60017)].