Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope
Abstract
Let d≥ 2, 0<α<1, and let Γk be the successive arrival times of a standard Poisson process on (0,∞). Given independent uniform directions εk∈ Sd−1, independent of (Γk), we consider the random countable stable zonotope Zα=⨁k=1∞Γk−1/α[0,εk]. For its set of extreme points ext Zα, we prove that almost surely dimH ext Zα=(d−1)α, and that the critical Hausdorff measure ℋ(d−1)α(ext Zα) 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.
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BibTeX
@misc{kukushkin2026hausdorff,
title = {{Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope}},
author = {Maksim Kukushkin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28004},
eprint = {2608.28004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28004},
}
RIS
TY - UNPB
TI - Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope
AU - Maksim Kukushkin
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.28004
UR - https://arxiv.org/abs/2608.28004
AB - Let d≥ 2, 0<α<1, and let Γ_k be the successive arrival times of a standard Poisson process on (0,∞). Given independent uniform directions ε_k∈ S^d−1, independent of (Γ_k), we consider the random countable stable zonotope Z_α=⨁_k=1^∞Γ_k^−1/α[0,ε_k]. For its set of extreme points ext Z_α, we prove that almost surely dim_H ext Z_α=(d−1)α, and that the critical Hausdorff measure ℋ^(d−1)α(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.
ER -