Remarks on random countable stable zonotopes

Abstract

In the paper we study random stable countable zonotopes in Rd, given by Zα=⨁k=1∞Γk−1α[0,εk],0<α<1, where ⊕ stands for Minkowski sum, Γk=∑1kτj, {τj,j≥1} are i.i.d. random variables with common standard exponential distribution, and {εk,k≥1} are i.i.d. random vectors in Rd with common distribution σ, concentrated on the unit sphere of Rd. The sequences (τk) and (εk) are supposed to be independent. We consider these random sets as elements of the space Kd of all compact convex subsets of Rd with the Hausdorff distance. The boundary of Zα has non-trivial Cantor-type structure, and, in the case d=2 and under mild condition on σ, we prove that the Hausdorff dimension of the set of extremal points of the boundary of Zα is equal to α.As a by-product we provide some kind of geometrical characterization of (non-random) countable zonotopes in the class of zonoids, which are limits of zonotopes.