[Paper Review] Poisson polyhedra in high dimensions

Julia Hoerrmann, Daniel Hug+2 more|arXiv (Cornell University)|Mar 31, 2014

Point processes and geometric inequalities34 references3 citations

TL;DR

This paper investigates the asymptotic geometry of the zero cell in a parametric class of random hyperplane tessellations in high dimensions, showing that under suitable dimension-dependent choices of the distance exponent and intensity, the normalized zero cell satisfies the hyperplane conjecture with overwhelming probability as dimension tends to infinity. The key contribution is a novel link between the f-vector of the zero cell and dual intrinsic volumes, enabling asymptotic analysis of face counts and isoperimetric ratios.

ABSTRACT

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the $f$-vector of the zero cell to certain dual intrinsic volumes.

Motivation & Objective

• To investigate the asymptotic behavior of the zero cell in a parametric family of random hyperplane tessellations as space dimension n → ∞.
• To determine conditions under which the zero cell satisfies the hyperplane conjecture in high dimensions.
• To establish connections between the f-vector of the zero cell and dual intrinsic volumes for asymptotic analysis.
• To analyze the high-dimensional limits of the expected number of faces and the isoperimetric ratio of the zero cell.

Proposed method

• Introduces a parametric class of isotropic Poisson hyperplane processes in R^n depending on a distance exponent r > 0 and intensity γ.
• Defines the zero cell Z₀ as the random polyhedron containing the origin in the resulting hyperplane tessellation.
• Derives new identities linking the f-vector (number of ℓ-dimensional faces) to dual intrinsic volumes of Z₀.
• Uses these identities to bound expected face counts via estimates on dual intrinsic volumes, leveraging Stirling’s formula and asymptotic analysis.
• Applies the bounds to study the asymptotic behavior of the expected number of faces fℓ(Z₀) and the isoperimetric ratio as n → ∞.
• Analyzes the normalized zero cell (rescaled to unit volume) to assess its compliance with the hyperplane conjecture.

Experimental results

Research questions

• RQ1 Under what conditions on the distance exponent r and intensity γ does the normalized zero cell satisfy the hyperplane conjecture with high probability in high dimensions?
• RQ2 How do the expected numbers of ℓ-dimensional faces of the zero cell behave asymptotically as the space dimension n tends to infinity?
• RQ3 What is the limiting behavior of the isoperimetric ratio of the zero cell in high dimensions?
• RQ4 How do the f-vector and dual intrinsic volumes of the zero cell relate in this parametric random tessellation model?
• RQ5 Can the hyperplane conjecture be verified probabilistically via random polyhedra constructions in high dimensions?

Key findings

• For r = b n^α with b > 0 and α > 1/2, the probability that the normalized zero cell satisfies the hyperplane conjecture tends to one as n → ∞.
• The asymptotic growth rate of the expected number of ℓ-dimensional faces fℓ(Z₀) is bounded by √(2πb) in the n-th root sense when α > 1.
• When α = 1, the lower bound on the n-th root of E[f_{n−ℓ}(Z₀)] is √(1 + b)(1 + 1/b)^{b/2}, which increases with b.
• For α > 1, the lower bound on n^{(1−α)/2} times the n-th root of E[f_{n−ℓ}(Z₀)] converges to √(be), indicating exponential growth in face count.
• The isoperimetric ratio of the normalized zero cell converges to √(2πb) as n → ∞, providing a quantitative measure of shape concentration.
• The f-vector of the zero cell is linked to dual intrinsic volumes via new identities, generalizing earlier results such as those of Schneider (r=1) and Efron (random convex hulls).