arXiv subjects

Gilles Bonnet

Publications and source records attributed to Gilles Bonnet.

16 recordsLinked to original sources

arXiv2026

Wasserstein stability of the zero cell of a Poisson hyperplane tessellation under directional perturbations

A stationary Poisson hyperplane process in $\mathbb{R}^d$ is characterized by an intensity parameter and an even probability measure on the unit sphere called the directional distribution. In this work, we investigate the stability of the zero cell, i.e., the random convex polytope of the induced hyperplane tessellation containing the origin, under perturbations of the directional distribution. Our results provide quantitative bounds establishing local H\"{o}lder continuity of the distribution of the zero cell with respect to Wasserstein metrics on the space of random convex bodies and probability distributions on the unit sphere. As an application, we establish stability bounds for density estimators constructed from Poisson hyperplane tessellations. In particular, we bound the expected total variation distance between the zero-cell-based estimated probability measures in terms of the Wasserstein distance between their directional distributions.

math.PR↗

arXiv2025

Degrees in the $\beta$- and $\beta'$-Delaunay graphs

We investigate the typical cells $\widehat{Z}$ and $\widehat{Z}^\prime$ of $\beta$- and $\beta'$-Voronoi tessellations in $\mathbb{R}^d$, establishing a Complementary Theorem which entails: 1) a gamma distribution of the $\Phi$-content (a suitable homogeneous functional) of the typical cell with $n$-facets; 2) the independence of this $\Phi$-content with the shape of the cell; 3) a practical integral representation of the distribution of $Z^{(\prime)}$. We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of $\beta$- and $\beta'$-Delaunay triangulations. For $\beta'$-Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For $\beta$-Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for $d=2$).

math.PR↗

arXiv2024

Concentration inequalities for Poisson $U$-statistics

In this article we obtain concentration inequalities for Poisson $U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying Poisson point process $\eta$. The main result are new concentration bounds of the form \[ \mathbb{P}(|F_m ( f , \eta) -\mathbb{E} F_m ( f , \eta)| \ge t)\leq 2\exp(-I(\gamma,t)), \] where $I(\gamma,t)$ is of optimal order in $t$, namely it satisfies $I(\gamma,t)=\Theta(t^{1\over m}\log t)$ as $t\to\infty$ and $\gamma$ is fixed. The function $I(\gamma,t)$ is given explicitly in terms of parameters of the assumptions satisfied by $f$ and $\Lambda$. One of the key ingredients of the proof is bounding the centred moments of $F_m(f,\eta)$. We discuss the optimality of obtained concentration bounds and consider a number of applications related to Gilbert graphs and Poisson hyperplane processes in constant curvature spaces.

math.PR↗

arXiv2023

Glass-like Caging with Random Planes

The richness of the mean-field solution of simple glasses leaves many of its features challenging to interpret. A minimal model that illuminates glass physics the same way the random energy model clarifies spin glass behavior would therefore be beneficial. Here, we propose such a real-space model that is amenable to infinite-dimensional analysis and is exactly solvable at high and low densities in finite dimension. By joining analysis with numerical simulations, we uncover geometrical signatures of the dynamical and jamming transitions and provides insight into the origin of activated processes. Translating these findings to the context of standard glass formers further reveals the role played by non-convexity in the emergence of Gardner physics.

cond-mat.dis-nn↗

arXiv2022

Limit theory of sparse random geometric graphs in high dimensions

We study topological and geometric functionals of $l_\infty$-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

math.PR↗

arXiv2021

Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $Ω(n m^{\frac{1}{n-1}})$ and $O(n^2 m^{\frac{1}{n-1}} + n^5 4^n)$ with high probability when $m \geq 2^{Ω(n)}$. For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when $m$ is large, where we rely on the $Θ(n^2 m^{\frac{1}{n-1}})$ bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these ``shadows paths'' together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope $P^\circ$, corresponding to a random convex hull, by showing the relation $\operatorname{diam}(P) \geq (n-1)(\operatorname{diam}(P^\circ)-2)$. We then prove that the shortest path between any ``nearly'' antipodal pair vertices of $P^\circ$ has length $Ω(m^{\frac{1}{n-1}})$.

math.PR↗

arXiv2021

Sharp inequalities for the mean distance of random points in convex bodies

For a convex body $K\subset\mathbb{R}^d$ the mean distance $Δ(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $Δ(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.

math.MG↗

arXiv2021

Phase transition for the volume of high-dimensional random polytopes

The beta polytope $P_{n,d}^β$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^β$ if $β>-1$ (in particular, $β=0$ corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if $β=-1$. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when $β=0$, their number of vertices.

math.PR↗

arXiv2020

Weak convergence of the intersection point process of Poisson hyperplanes

This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distance is provided as well. In the background is a general functional Poisson approximation theorem on abstract Poisson spaces. Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its $f$-vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14.3 (1993), 381--394] in computational geometry.

math.PR↗

arXiv2019

Facets of spherical random polytopes

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for $n$ and $d$ with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of some known results in fixed dimension to the case where dimension tends to infinity are described.

math.PR↗

arXiv2018

Small cells in a Poisson hyperplane tessellation

Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size functionals and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $φ$ and with respect to an arbitrary size functional $Σ$. More precisely, we investigate the distribution of the typical cell $Z$, conditioned on the event $\{Σ(Z) 0$, and increasing with respect to set inclusion. We focus on the number of facets and the shape of such small cells. We show in various general settings that small cells tend to minimize the number of facets and that they have a non degenerated limit shape distribution which depends on the size $Σ$ and the directional distribution. We also exhibit a class of directional distribution for which cells with small inradius do not tend to minimize the number of facets.

math.MG↗

arXiv2018

Threshold phenomena for high-dimensional random polytopes

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

math.MG↗

arXiv2018

The maximal degree in a Poisson-Delaunay graph

We investigate the maximal degree in a Poisson-Delaunay graph in $\mathbf{R}^d$, $d\geq 2$, over all nodes in the window $\mathbf{W}_ρ:= ρ^{1/d}[0,1]^d$ as $ρ$ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when $d\geq 3$

math.PR↗

arXiv2017

Monotonicity of facet numbers of random convex hulls

Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.

math.MG↗

arXiv2016

Polytopal approximation of elongated convex bodies

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body $K$ by a circumscribed polytope $P$ with a given number of facets. These bounds are of particular interest if $K$ is elongated. To measure the elongation of the convex set, its isoperimetric ratio $ V_j(K)^{1/j} V_i(K)^{-1/i} $ is used.

math.MG↗

arXiv2016

Cells with many facets in a Poisson hyperplane tessellation

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 $Φ$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $Φ$ content is large.

math.PR↗