Asymptotics for some combinatorial characteristics of the convex hull of a Poisson point process in the Clifford torus

Abstract

N. Dolbilin and M. Tanemura studied the convex hulls of finite subsets of the Clifford torus $T$ in $E^4$. They have completely studied the combinatorial structure of the convex hull for a periodic point set. Moreover, there was performed a numerical simulation of the convex hull for the Poisson point process on $T$ that showed that the mean valence of a vertex of the convex hull has asymptotics $O^*(\ln \lambda)$ where $\lambda$ is the rate of the process. N. Dolbilin suggested the author to prove the conjecture on the logarithmic growth of the mean degree of a vertex. In this paper we prove this conjecture and some related theorems.