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

Abstract: Let $d\geq 2$, $0<α<1$, and let $Γ<em>k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S<sup>{d-1}$, independent of $(Γ_k)$, we consider the random countable stable zonotope $$Z</em>α=\bigoplus_{k=1}<sup>{\infty}Γ_k<sup>{-1/α}[0,\varepsilon_k]$$. 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.