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Register a new userConvex hulls of perturbed random point sets
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We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^{alpha}, alpha in (-infty, infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $alpha$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $alpha = frac{-2} {d -1}$ and $alpha = frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.
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Let $U_1,U_2,ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $ntoinfty$, 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, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. 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 $|x|^{-(d+gamma)}$, 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 $gamma>0$.
Let $xi_1,xi_2,ldots$ be a sequence of independent copies of a random vector in $mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=xi_1+cdots+xi_i$, and let $C_{n,d}:=text{conv}(0,S_1,S_2,ldots,S_n)$ be the convex hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0leq i_0 <cdots < i_kleq n$ the convex hull of the $k+1$ points $S_{i_0},ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic regimes, i.e. when $n$, $d$, and possibly also $k$ diverge to $infty$. There is an explicit formula for the expected number of $k$-dimensional faces of $C_{n,d}$ which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of $C_{n,d}$. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.
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 $X_1,ldots,X_n$ ($ngeq d+1$). 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 $n$.
Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $mathbb{S}^{d}subsetmathbb{R}^{d+1}$ with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as $N to infty$) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of $N$ random points on $mathbb{S}^{d}$. We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on $mathbb{S}^{d}$.
We consider a variant of a classical coverage process, the boolean model in $mathbb{R}^d$. Previous efforts have focused on convergence of the unoccupied region containing the origin to a well studied limit $C$. We study the intersection of sets centered at points of a Poisson point process confined to the unit ball. Using a coupling between the intersection model and the original boolean model, we show that the scaled intersection converges weakly to the same limit $C$. Along the way, we present some tools for studying statistics of a class of intersection models.
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