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

TL;DR

This paper proves a conjecture that the average degree of vertices in the convex hull of a Poisson point process on the Clifford torus grows logarithmically with the process rate, confirming numerical and theoretical predictions.

Contribution

The paper establishes the logarithmic asymptotic growth of the mean vertex degree in the convex hull of a Poisson process on the Clifford torus, confirming a conjecture and extending prior numerical findings.

Findings

Mean valence of a vertex grows as O*(ln λ)

Proved conjecture on logarithmic growth

Extended understanding of convex hulls in toroidal geometries

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.