Samenvatting
Let Z be the typical cell of a stationary Poisson hyperplane
tessellation in Rd. The distribution of the number of facets
f(Z) of the typical cell is investigated. It is shown, that under
a well-spread condition on the directional distribution, the
quantity n 2
d−1
n P(f(Z) = n) is bounded from above and from
below. When f(Z) is large, the isoperimetric ratio of Z is
bounded away from zero with high probability.
These results rely on one hand on the Complementary
Theorem which provides a precise decomposition of the
distribution of Z and on the other hand on several geometric
estimates related to the approximation of polytopes by
polytopes with fewer facets.
From the asymptotics of the distribution of f(Z), tail
estimates for the so-called Φ content of Z are derived as well
as results on the conditional distribution of Z when its Φ
content is large.
tessellation in Rd. The distribution of the number of facets
f(Z) of the typical cell is investigated. It is shown, that under
a well-spread condition on the directional distribution, the
quantity n 2
d−1
n P(f(Z) = n) is bounded from above and from
below. When f(Z) is large, the isoperimetric ratio of Z is
bounded away from zero with high probability.
These results rely on one hand on the Complementary
Theorem which provides a precise decomposition of the
distribution of Z and on the other hand on several geometric
estimates related to the approximation of polytopes by
polytopes with fewer facets.
From the asymptotics of the distribution of f(Z), tail
estimates for the so-called Φ content of Z are derived as well
as results on the conditional distribution of Z when its Φ
content is large.
Originele taal-2 | English |
---|---|
Pagina's (van-tot) | 203-240 |
Aantal pagina's | 37 |
Tijdschrift | Advances in Mathematics |
Volume | 324 |
DOI's | |
Status | Published - 14-jan.-2018 |
Extern gepubliceerd | Ja |