Geometrical characteristics of random countable zonotopes

M. A. Kukushkin

Saint Petersburg State University

Abstract: This note studies random countable zonotopes in $\mathbb{R}^d$, i.e., objects of the form $Z_\alpha = \bigoplus_{k=1}^{\infty}\Gamma_k^{-\frac{1}{\alpha}}[0,\varepsilon_k], 0 < \alpha < 1$, where $\Gamma_k=\sum_{j=1}^{k}\tau_j$, $\{\tau_j, j\ge 1\}$ are i.i.d. random variables with a common standard exponential distribution, and $\{ \varepsilon_j, j\ge 1\}$ are i.i.d. random vectors with a common distribution $\sigma$ concentrated on the unit sphere $S^{d-1}\subset \mathbb{R}^d$. The sequences $\{\tau_j\}$ and $\{\varepsilon_j\}$ are assumed to be independent. The boundary of such a centrally symmetric body, $\partial Z_\alpha$, has a non-trivial Cantor-like structure. Of interest is the estimation of the Hausdorff dimension of the set of its extreme points, $\dim_H(\mathrm{ext}\,{Z_\alpha})$. For dimension $d=2$, upper and lower bounds were obtained in Davydov and Paulauskas (2019). However, for $d=3$ and in higher dimensions, the question of both the lower and the upper bound remained open. Below, we state and prove a new result concerning the upper bound in an arbitrary dimension. Namely, we claim that the inequality $\dim_H(\mathrm{ext}\,{Z_\alpha}) \le \alpha + d - 2$ holds almost surely for any $d \ge 2$ and $0 < \alpha < 1$.

Key words and phrases: stable random variables, zonotopes, random countable zonotopes, abstract cones, Hausdorff dimension, extreme points.

UDC: 519.2

Received: 14.10.2025