Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain
Let \(A \subseteq \mathbb{R}^2\) be an unbounded convex set such that for all \(c \in \mathbb{R}\) the set \(A \cap \{(x,y) \mid y \leq c\}\) has finite area and let \(C(A)\) denote the convex hull generated by the restriction of a homogeneous Poisson point process in \(\mathbb{R}^2\) to the set \(A\). As the main result of this paper it is shown that the set \(A \backslash C(A)\) can be represented as the union of curvilinear triangles with independent, exponentially distributed areas. It is also indicated how to simulate \(C(A)\) directly without first simulating the process itself. In the case when \(A\) is a cone the properties of this representations are examined more completely.
scientific article; zbMATH DE number 786370