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

Abstract

Let $d\ge 2$, $0<\alpha <1$, and let ${\Gamma }_k$ be the successive arrival times of a standard Poisson process on $(0,\infty )$. Given independent uniform directions ${\epsilon }_k\in S^{d-1}$, independent of $({\Gamma }_k)$, we consider the random countable stable zonotope $Z_{\alpha }={⨁}_{k=1}^{\infty }{\Gamma }_k^{-1/\alpha }[0,{\epsilon }_k]$. For its set of extreme points $\mathrm{ext}{Z}_{\alpha }$, we prove that almost surely ${\dim }_H\mathrm{ext}{Z}_{\alpha }=(d-1)\alpha$, and that the critical Hausdorff measure ${\mathcal{H}}^{(d-1)\alpha }(\mathrm{ext}{Z}_{\alpha })$ 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.

20 pages, 4 figures. Added a reference and discussion of prior work; results unchanged