Let $μ$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $μ$. We establish a sharp threshold for the product measure $μ_n$ of the random polytope $K_N:={\rm conv}\bigl\{\vec{X}_1,\ldots,\vec{X}_N\bigr\}$ in ${\mathbb R}^n$ under the assumption that the Legendre transform $Λ_μ^{\ast}$ of the logarithmic moment generating function of $μ$ satisfies the condition $$\lim\limits_{x\uparrow x^{\ast}}\dfrac{-\ln μ([x,\infty ))}{Λ_μ^{\ast}(x)}=1,$$ where $x^{\ast}=\sup\{x\in\mathbb{R}\colon μ([x,\infty))>0\}$. An application is a sharp threshold for the case of the product measure $ν_p^n=ν_p^{\otimes n}$, $p\geq 1$ with density $(2γ_p)^{-n}\exp(-\|x\|_p^p)$, where $\|\cdot\|_p$ is the $\ell_p^n$-norm and $γ_p=Γ(1+1/p)$.
As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decompositions and Choquet convex decompositions, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull. Furthermore, we show that Choquet decomposable hull of a set coincides with its (strongly) closed …
We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the $d$-dimensional Gaussian polytope with or without the origin.
We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.
In Cabo and Groeneboom (1994) the remaining area of the left-lower convex hull of a Poisson point process with intensity one in the first quadrant of the plane was analyzed, using the methods of Groeneboom (1988), giving formulas for the expectation and variance of the remaining area for a finite interval of slopes of the boundary of the convex hull. However, the time inversion argument of Groeneboom (1988) was not correctly applied in Cabo and Groeneboom (1994), leading to an incorrect scaling constant for the variance. The purpose of this note is to show how the correct application of the ti…
We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.
The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number of k-dimensional faces and the outer defect intrinsic volumes of the first layers of the convex hull peeling of a homogeneous Poisson point process in the unit ball whose intensity goes to infinity. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, the growth rates do not depend …
We investigate measures of complexity of function classes based on continuity moduli of Gaussian and Rademacher processes. For Gaussian processes, we obtain bounds on the continuity modulus on the convex hull of a function class in terms of the same quantity for the class itself. We also obtain new bounds on generalization error in terms of localized Rademacher complexities. This allows us to prove new results about generalization performance for convex hulls in terms of characteristics of the base class. As a byproduct, we obtain a simple proof of some of the known bounds on the entropy of co…
It is well known that for standard Brownian motion $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$ its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability 1 contains 0 as an interior point for each $t > 0$ (see \cite{E}). The aim of this note is to state the analoguos property for $d$-dimensional fractional Brownian motion.
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 $n$.
We introduce planar random walk conditioned to avoid its past convex hull, and we show that it escapes at a positive limsup speed. Experimental results show that fluctuations from a limiting direction are on the order of n^(3/4). This behavior is also observed for the extremal investor, a natural financial model related to the planar walk.
The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are derived. This is one of the first investigations leading to rigorous results for random polytopes which are neither simple nor simplicial. The results contrast existing results when points are chosen in the interior of a convex set.
In this paper we consider the convex hull of a spherically symmetric sample in $R^d$. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or $O$-regularly varying.
We consider a random walk in the plane which takes steps uniformly distributed on the unit circle centered around the walker's current position but avoids the convex hull of its past positions. This model has been introduced by Angel, Benjamini and Virag. We show a large deviation estimate for the distance of the walker from the origin, which implies that the walker has positive lim inf speed.
We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf_{n \to \infty} L_n/D_n =2$ and $\limsup_{n \to \infty} L_n /D_n = …
We establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent Brownian paths in the plane. This requires the introduction of a counting criterion which amounts to "cutting off" edges that are, in a specific sense, small. The main argument consists in a mapping between planar Brownian convex hulls and configurations of constrained, independent linear Brownian motions. This new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of Brownian convex hulls in the plane.
Let $X(t)$, $t\geq0$, be a Lévy process in $\mathbb{R}^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set $Z_s$ and find explicit expressions for their means in the case of symmetric $α$-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit theorems for the convex hull of Lévy processes with normal and stable lim…
It is well known that for a standard Brownian motion (BM) $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$, its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability $1$ for each $t > 0$ contains $0$ as an interior point (see Evans (1985)). We also know that the winding number of a typical path of a $2$-dimensional BM is equal to $+\infty.$ The aim of this article is to show that these properties aren't specifically "Brownian", but hold for a much larger class of $d$-dimensional self similar processes. This class contains in particular $d$-dimensional fractional Brownian mot…
For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the c…
We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation Lévy processes and depends subtly on the behaviour of the Lévy measure at zero. We introduce a class of strongly eroded Lévy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of…
The paper reexamines an argument by Talagrand that leads to a remarkable exponential tail bound for the concentration of probability near a set. The main novelty is the replacement of a mysterious Calculus inequality by an application of Jensen's inequality.
Let alpha \in (1, 2] and X be an R^d-valued alpha-stable process with independent and symmetric components starting in 0. We consider the closure S_t of the path described by X on the interval [0, t] and its convex hull Z_t. The first result of this paper provides a formula for certain mean mixed volumes of Z_t and in particular for the expected first intrinsic volume of Z_t. The second result deals with the asymptotics of the expected volume of the stable sausage Z_t+B (where B is an arbitrary convex body with interior points) as t \to 0.
We give new proofs of the description convex hulls of space curves $γ: [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to $γ$.
We prove that the convex peeling of a random point set in dimension d approximates motion by the 1/(d + 1) power of Gaussian curvature. We use viscosity solution theory to interpret the limiting partial differential equation. We use the Martingale method to solve the cell problem associated to convex peeling. Our proof follows the program of Armstrong-Cardaliaguet for homogenization of geometric motions, but with completely different ingredients.
