Random Polytopes

2009 ◽  
pp. 45-76 ◽  
Author(s):  
Matthias Reitzner
Keyword(s):  
Random Polytopes

On the Isotropic Constant of Random Polytopes with Vertices on an $$\ell _p$$ ℓ p -Sphere

2017 ◽  
Vol 28 (1) ◽  
pp. 405-426 ◽  
Author(s):  
Julia Hörrmann ◽  
Joscha Prochno ◽  
Christoph Thäle
Keyword(s):  
Random Polytopes ◽  
Isotropic Constant

scholarly journals Random polytopes: Central limit theorems for intrinsic volumes

2018 ◽  
Vol 146 (7) ◽  
pp. 3063-3071 ◽  
Author(s):  
Christoph Thäle ◽  
Nicola Turchi ◽  
Florian Wespi
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Random Polytopes ◽  
Intrinsic Volumes

scholarly journals Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

2014 ◽  
Vol 46 (4) ◽  
pp. 919-936
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Hausdorff Distance ◽  
Dimensional Euclidean Space ◽  
Approximation Properties ◽  
Random Polytopes ◽  
Directional Distribution ◽  
Zero Cell

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

scholarly journals Stability Properties of Neighbourly Random Polytopes

2008 ◽  
Vol 41 (2) ◽  
pp. 257-272 ◽  
Author(s):  
Piotr Mankiewicz ◽  
Nicole Tomczak-Jaegermann
Keyword(s):  
Random Polytopes ◽  
Stability Properties

scholarly journals Large deviation probabilities for the number of vertices of random polytopes in the ball

2006 ◽  
Vol 38 (01) ◽  
pp. 47-58 ◽  
Author(s):  
Pierre Calka ◽  
Tomasz Schreiber
Keyword(s):  
Unit Ball ◽  
Point Process ◽  
Poisson Point Process ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Extreme Points ◽  
Auxiliary Result ◽  
Random Polytopes ◽  
Large Numbers ◽  
Strong Localization

In this paper we establish large deviation results on the number of extreme points of a homogeneous Poisson point process in the unit ball of R d . In particular, we deduce an almost-sure law of large numbers in any dimension. As an auxiliary result we prove strong localization of the extreme points in an annulus near the boundary of the ball.

scholarly journals Random polytopes: Their definition, generation and aggregate properties

1982 ◽  
Vol 24 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Jerrold H. May ◽  
Robert L. Smith
Keyword(s):  
Random Polytopes ◽  
Aggregate Properties

scholarly journals A new approach to weak convergence of random cones and polytopes

2020 ◽  
pp. 1-21
Author(s):  
Zakhar Kabluchko ◽  
Daniel Temesvari ◽  
Christoph Thäle
Keyword(s):  
Weak Convergence ◽  
Convex Sets ◽  
Compact Convex ◽  
New Approach ◽  
Random Polytopes ◽  
Typical Cell

Abstract A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated by n independent and uniformly distributed random linear hyperplanes in $\mathbb {R}^{d+1}$ , weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb {R}^d$ , as $n\to \infty $ .

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