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

Abstract

Let $d\geq 2$, $0<\alpha<1$, and let $\Gamma_k$ 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 $(\Gamma_k)$, we consider the random countable stable zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\alpha}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_\alpha$, we prove that almost surely $\dim_H \operatorname{ext} Z_\alpha=(d-1)\alpha$, and that the critical Hausdorff measure $\mathcal H^{(d-1)\alpha}(\operatorname{ext} Z_\alpha)$ 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.