Convex bodies: the brunn–minkowski theory

Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,β(x)=const·1−∥x∥2β,∥x∥

Expected intrinsic volumes and facet numbers of random beta-polytopes

Let $r=r(n)$ be a sequence of integers such that $r\leq n$ and let $X_1,\ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $\mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,\ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$\phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $r\sim \alpha n$ for some $0 < \alpha < 1$).

Limit theorems for random simplices in high dimensions

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin–Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^{d}$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

/pdf/normal-approximation-for-stabilizing-functionals-2j6lzyj6kf.pdf

Normal approximation for stabilizing functionals

Let $$U_1,U_2,\ldots $$
 be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $$n\rightarrow \infty $$
 , the f-vector of the $$(d+1)$$
 -dimensional convex cone $$C_n$$
 generated by $$U_1,\ldots ,U_n$$
 weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of $$C_n$$
 and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $$C_n$$
 can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644
 ). Our approach is based on the observation that the random cone $$C_n$$
 weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $$\Vert x\Vert ^{-(d+\gamma )}$$
 , where $$\gamma =1$$
 . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary $$\gamma >0$$
 .

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Monotonicity of facet numbers of random convex hulls

Let X1,…,XN, N > n, be independent random points in ℝn, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrin...

Threshold phenomena for high-dimensional random polytopes

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to a class of zero cells associated with Poisson hyperplane mosaics. As a special case, this comprises the typical Poisson–Voronoi cell conditioned on having large inradius.

Concentration and moderate deviations for Poisson polytopes and polyhedra

Let $K$ be a convex body in $\mathbb{R} ^n$ and $f : \partial K \rightarrow \mathbb{R} _+$ a continuous, strictly positive function with $\int \limits _{\partial K} f(x) \mathrm{d} \mu _{\partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $\mathbb{R} ^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner [36]. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.

/pdf/approximation-of-smooth-convex-bodies-by-random-polytopes-3w3anuly89.pdf

Julian Grote

Papers

Limit theorems for random simplices in high dimensions

Monotonicity of facet numbers of random convex hulls

Threshold phenomena for high-dimensional random polytopes

Concentration and moderate deviations for Poisson polytopes and polyhedra

Approximation of smooth convex bodies by random polytopes