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Register a new userOn the cells in a stationary Poisson hyperplane mosaic
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Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $hat X$ in ${mathbb R}^d$. Under some mild conditions on the spherical directional distribution of $hat X$ (which are satisfied, for example, if the process is isotropic), we show that with probability one the set of cells ($d$-polytopes) of $X$ has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple $d$-polytopes is realized infinitely often by the cells of $X$. A further result concerns the distribution of the typical cell.
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Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $mathbb{R}^d$. 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 $n^{frac{2}{d-1}}sqrt[n]{mathbb{P}(f(Z)=n)}$ 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 $Phi$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $Phi$ content is large.
A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the non-negative side of the arrangement ${xin R^d : p_{H}(x)ge 0}$ is the same as the measure on the non-positive side ${xin R^d : p_{H}(x)le 0}$. In 2017 Barba, Pilz & Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $dge lceil j/k rceil$. The conjecture was confirmed in the case when $dge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevic, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grunbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+ell$ measures on $R^{2^a+ell}$, where $1leq ellleq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.
We use symplectic techniques to obtain partial results on Mahlers conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $ell_p$-balls or the Hanner polytopes.
We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(alpha,0)$ and $(alpha,alpha)$. The construction has two steps. The first is a general construction of interval partition processes obtained previously, by decorating the jumps of a Levy process with independent excursions. Here, we focus on the second step, which requires explicit transition kernels and what we call pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper, we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards resolving longstanding conjectures by Feng and Sun on measure-valued Poisson-Dirichlet diffusions, and by Aldous on a continuum-tree-valued diffusion.
Let $[mathcal{P}]$ be the points of a Poisson process on $mathbb{R}^d$ and $F$ a probability distribution with support on the non-negative integers. Models are formulated for generating translation invariant random graphs with vertex set $[mathcal{P}]$ and iid vertex degrees with distribution $F$, and the length of the edges is analyzed. The main result is that finite mean for the total edge length per vertex is possible if and only if $F$ has finite moment of order $(d+1)/d$.
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