Random Polytopes Research Articles

On the variance of the mean width of random polytopes circumscribed around a convex body

AbstractLet be a convex body in in which a ball rolls freely and which slides freely in a ball. Let be the intersection of i.i.d. random half‐spaces containing chosen according to a certain prescribed probability distribution. We prove an asymptotic upper bound on the variance of the mean width of as . We achieve this result by first proving an asymptotic upper bound on the variance of the weighted volume of random polytopes generated by i.i.d. random points selected according to certain probability distributions, then, using polarity, we transfer this to the circumscribed model.

Random polytopes generated by contoured distributions

Our work is a further investigation on the connection between probability and geometry. We extend several known results from convex bodies and log-concave measures to the setting of contoured probability distributions as we study the relation of several parameters of a convex body, a profile function and the resulting contoured distribution, where log-concavity is not required.

Stochastic Gain-Scheduled Control of Discrete-Time Systems Characterized by Random Polytopes

Stochastic Gain-Scheduled Control of Discrete-Time Systems Characterized by Random Polytopes

Half-space depth of log-concave probability measures

Given a probability measure mu on {{mathbb {R}}}^n, Tukey’s half-space depth is defined for any xin {{mathbb {R}}}^n by varphi _{mu }(x)=inf {mu (H):Hin {{{mathcal {H}}}}(x)}, where mathcal{H}(x) is the set of all half-spaces H of {{mathbb {R}}}^n containing x. We show that if mu is a non-degenerate log-concave probability measure on {{mathbb {R}}}^n then e-c1n⩽∫Rnφμ(x)dμ(x)⩽e-c2n/Lμ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} e^{-c_1n}\\leqslant \\int _{{\\mathbb {R}}^n}\\varphi _{\\mu }(x)\\,d\\mu (x) \\leqslant e^{-c_2n/L_{\\mu }^2} \\end{aligned}$$\\end{document}where L_{mu } is the isotropic constant of mu and c_1,c_2>0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of L_q-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

Asymptotic expected T‐functionals of random polytopes with applications to Lp$L_p$ surface areas

AbstractAn asymptotic formula is proved for the expected T‐functional of the convex hull of independent and identically distributed random points sampled from the Euclidean unit sphere in according to an arbitrary positive continuous density. As an application, the approximation of the sphere by random polytopes in terms of surface area differences is discussed.

Threshold for the expected measure of the convex hull of random points with independent coordinates

AbstractLet be an even Borel probability measure on . For every , consider independent random vectors in , with independent coordinates having distribution . We establish a sharp threshold for the product measure of the random polytope in under the assumption that the Legendre transform of the logarithmic moment generating function of satisfies the condition where . An application is a sharp threshold for the case of the product measure , with density , where is the ‐norm and .

Spherical convex hull of random points on a wedge

Consider two half-spaces H1+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_1^+$$\\end{document} and H2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_2^+$$\\end{document} in Rd+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{d+1}$$\\end{document} whose bounding hyperplanes H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_1$$\\end{document} and H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_2$$\\end{document} are orthogonal and pass through the origin. The intersection S2,+d:=Sd∩H1+∩H2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}_{2,+}^d:={\\mathbb {S}}^d\\cap H_1^+\\cap H_2^+$$\\end{document} is a spherical convex subset of the d-dimensional unit sphere Sd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^d$$\\end{document}, which contains a great subsphere of dimension d-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d-2$$\\end{document} and is called a spherical wedge. Choose n independent random points uniformly at random on S2,+d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}_{2,+}^d$$\\end{document} and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log n$$\\end{document}. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on S2,+d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}_{2,+}^d$$\\end{document}. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

Angle Sums of Random Polytopes

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

Random Zonotopes and Valuations

We define a random zonotope in $${\mathbb R}^d$$ , by adding finitely many random segments, which are independently and identically distributed. For this random polytope, we determine, under a mild assumption on the distribution, the expectations of the intrinsic volumes, more generally, the expectations of suitable valuations. We also prove a central limit theorem for a valuation evaluated at these random zonotopes.

Threshold for the expected measure of random polytopes

Let μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} be a log-concave probability measure on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^n$$\\end{document} and for any N>n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N>n$$\\end{document} consider the random polytope KN=conv{X1,…,XN}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N=\ extrm{conv}\\{X_1,\\ldots ,X_N\\}$$\\end{document}, where X1,X2,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_1,X_2,\\ldots $$\\end{document} are independent random points in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^n$$\\end{document} distributed according to μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. We study the question if there exists a threshold for the expected measure of KN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N$$\\end{document}. Our approach is based on the Cramer transform Λμ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _{\\mu }^{*}$$\\end{document} of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. We examine the existence of moments of all orders for Λμ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _{\\mu }^{*}$$\\end{document} and establish, under some conditions, a sharp threshold for the expectation EμN[μ(KN)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}_{\\mu ^N}[\\mu (K_N)]$$\\end{document} of the measure of KN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N$$\\end{document}: it is close to 0 if lnN≪Eμ(Λμ∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln N\\ll {\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*})$$\\end{document} and close to 1 if lnN≫Eμ(Λμ∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln N\\gg {\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*})$$\\end{document}. The main condition is that the parameter β(μ)=Varμ(Λμ∗)/(Eμ(Λμ∗))2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (\\mu )=\ extrm{Var}_{\\mu }(\\Lambda _{\\mu }^{*})/({\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*}))^2$$\\end{document} should be small.

The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

An identity due to Efron dating from 1965 relates the expected volume of the convex hull of n random points to the expected number of vertices of the convex hull of n+1 random points. Forty years later this identity was extended from expected values to higher moments. The generalized identity has attracted considerable interest. Whereas the left-hand side of the generalized identity—concerning the volume—has an immediate geometric interpretation, this is not the case for the right-hand side—concerning the number of vertices. A transformation of the right-hand side applying an identity for elementary symmetric polynomials overcomes the blemish. The arising formula reveals a duality between the volumes and the numbers of vertices of random polytopes.

The Convex Hull of Random Points on the Boundary of a Simple Polytope

The convex hull of N independent random points chosen on the boundary of a simple polytope in {mathbb {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.

Robust H2 Analysis of Discrete-Time Linear Systems Characterized by Random Polytopes and Time-Varying Parameters

This paper deals with discrete-time linear stochastic systems characterized by random polytopes and time-varying parameters. Random polytopes and time-varying parameters are introduced so that they can represent a sort of temporal variations in the distributions of coefficient random matrices of the systems. An H2 norm is defined for associated stochastic systems, and H2 performance analysis is discussed. In particular, we derive a numerically tractable linear matrix inequality condition for such analysis, as an extension of our earlier results about H2 control for stochastic systems without random polytopes.

Facets of spherical random polytopes

AbstractFacets of the convex hull of n independent random vectors chosen uniformly at random from the unit sphere in 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 several known results in fixed dimension to the case where dimension tends to infinity are described.

On Wendel’s equality for intersections of balls

We study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space mathbb {R}^{d} that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for nge d+1 points in mathbb {R}^{d}, for the spindle convex hull this happens for all nge 2. We study this question for the uniform and normally distributed random models.

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

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 \(C_{n,d}:={{\,\mathrm{conv}\,}}(0,S_1,S_2,\ldots ,S_n)\) be the convex hull of the first \(n+1\) points it has visited. The polytope \(C_{n,d}\) is called k-neighborly if for any indices \(0\le i_1<\cdots < i_k\le n\) the convex hull of the k points \(S_{i_1},\ldots , S_{i_k}\) is a \((k-1)\)-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 \((k-1)\)-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.

Convex cones spanned by regular polytopes

Abstract We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

On expected face numbers of random beta and beta’ polytopes

The random beta polytope is defined as the convex hull of n independent random points with the density proportional to (1-Vert xVert ^2)^beta on the d-dimensional unit ball, where beta >-1 is a parameter. Similarly, the random beta’ polytope is defined as the convex hull of n independent random points with the density proportional to (1+Vert xVert ^2)^{-beta } on mathbb R^d, where beta >frac{d}{2}. In a previous work (Kabluchko, Adv. Math. 380:107612, 2021), we established exact and explicit formulae for the expected f-vectors of these random polytopes in terms of certain definite integrals. In the present paper, we use purely algebraic manipulations to derive several identities for these integrals which yield alternative formulae for the expected f-vectors. Similar algebraic manipulations apply to Stirling numbers and yield the following identity: ∑s=0kn-sd-s(d-s)d-sk-s=∑s=0k(-1)sn-sdd+1k-s=∑s=0d-k(-1)sn+1d-sd-sk.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{s=0}^k \\genfrac\\rbrace \\lbrace {0.0pt}{}{n-s}{d-s} (d-s) \\genfrac[]{0.0pt}{}{d-s}{k-s}= & {} \\sum _{s=0}^k (-1)^s \\genfrac\\rbrace \\lbrace {0.0pt}{}{n-s}{d} \\genfrac[]{0.0pt}{}{d+1}{k-s} \\\\= & {} \\sum _{s=0}^{d-k} (-1)^s \\genfrac\\rbrace \\lbrace {0.0pt}{}{n+1}{d-s} \\genfrac[]{0.0pt}{}{d-s}{k}. \\end{aligned}$$\\end{document}

Doubly random polytopes

AbstractA two‐step model for generating random polytopes is considered. For parameters d, m, and p, the first step is to generate a simple polytope P whose facets are given by m uniform random hyperplanes tangent to the unit sphere in , and the second step is to sample each vertex of P independently with probability p and let Q be the convex hull of the sampled vertices. We establish results on how well Q approximates the unit sphere in terms of m and p as well as asymptotics on the combinatorial complexity of Q for certain regimes of p.

The Beauty of Random Polytopes Inscribed in the 2-Sphere

Consider a random set of points on the unit sphere in , which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.