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PreprintCC BY 4.0arXiv

The distributions of convex body functionals of the Poisson polyhedron

  • Felix Ballani
  • Dietrich Stoyan
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Abstract

This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.

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Cite

BibTeXRIS
BibTeX
@misc{ballani2026distributions,
  title = {{The distributions of convex body functionals of the Poisson polyhedron}},
  author = {Felix Ballani and Dietrich Stoyan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12514},
  eprint = {2609.12514},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12514},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The distributions of convex body functionals of the Poisson polyhedron
AU  - Felix Ballani
AU  - Dietrich Stoyan
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12514
UR  - https://arxiv.org/abs/2609.12514
AB  - This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.
ER  -