Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope

Source abstract

Let $d\geq 2$, $0<α<1$, and let $Γ_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(Γ_k)$, we consider the random countable stable zonotope $Z_α=\bigoplus_{k=1}^{\infty}Γ_k^{-1/α}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_α$, we prove that almost surely $\dim_H \operatorname{ext} Z_α=(d-1)α$, and that the critical Hausdorff measure $\mathcal H^{(d-1)α}(\operatorname{ext} Z_α)$ is almost surely finite. The lower bound follows from the tangential non-degeneracy of the stable increments of the parametrizing field and Frostman's energy criterion. For the upper bound we construct an adaptive covering: at each scale the Poisson jumps are split into large and small ones, the large jumps determine a finite hyperplane arrangement, and the sum of the small jumps controls the diameters of the images of its cells.