Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T04:24:53.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Random affine simplexes

Part of: Geometric probability and stochastic geometry General convexity Probability theory on algebraic and topological structures Optics, electromagnetic theory - General

Published online by Cambridge University Press:  12 July 2019

Friedrich Götze ,
Anna Gusakova  and
Dmitry Zaporozhets
Friedrich Götze*
Affiliation:
Bielefeld University
Anna Gusakova*
Affiliation:
Bielefeld University
Dmitry Zaporozhets*
Affiliation:
St Petersburg Department of the Steklov Mathematical Institute
*
*Postal address: Faculty of Mathematics, Bielefeld University, PO box 10 01 31, 33501 Bielefeld, Germany. Email address: [email protected]; [email protected]
*Postal address: Faculty of Mathematics, Bielefeld University, PO box 10 01 31, 33501 Bielefeld, Germany. Email address: [email protected]; [email protected]
**Postal address: St Petersburg Department of the SteklovMathematical Institute, Fontanka 27, 191023 St Petersburg, Russia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

Keywords

Blaschke–Petkantschin formulaconvex hullellipsoidexpected volumeFurstenberg–Tzkoni formulaGaussian matrixintrinsic volumerandom sectionrandom simplex

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 78A40: Waves and radiation 52A39: Mixed volumes and related topics
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 39 - 51
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bäsel, U. (2014). Random chords and point distances in regular polygons. Acta Math. Univ. Comenianae 83, 1–18.Google Scholar
Borel, É. (1925). Principes et Formules Classiques du Calcul des Probabilités. Gauthier-Villars, Paris.Google Scholar
Chakerian, G. (1967). Inequalities for the difference body of a convex body. Proc. Amer. Math. Soc. 18, 879–884.CrossRefGoogle Scholar
Dafnis, N. and Paouris, G. (2012). Estimates for the affine and dual affine quermassintegrals of convex bodies. Illinois J. Math. 56, 1005–1021.CrossRefGoogle Scholar
Furstenberg, H. and Tzkoni, I. (1971). Spherical functions and integral geometry. Israel J. Math. 10, 327–338.CrossRefGoogle Scholar
Ghosh, B. (1951). Random distances within a rectangle and between two rectangles. Bull. Calcutta Math. Soc. 43, 17–24.Google Scholar
Grote, J., Kabluchko, Z. and Thäle, C. (2017). Limit theorems for random simplices in high dimensions. Preprint. Available at https://arxiv.org/abs/1708.00471.Google Scholar
Hansen, J. and Reitzner, M. (2004). Electromagnetic wave propagation and inequalities for moments of chord lengths. Adv. Appl. Prob. 36, 987–995.CrossRefGoogle Scholar
Heinrich, L. (2014). Lower and upper bounds for chord power integrals of ellipsoids. App. Math. Sci. 8, 8257–8269.Google Scholar
Kabluchko, Z., Temesvari, D. and Thäle, C. (2017). Expected intrinsic volumes and facet numbers of random beta-polytopes. Preprint. Available at https://arxiv.org/abs/1707.02253.Google Scholar
Kabluchko, Z. and Zaporozhets, D. (2014). Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields. J. Math. Sci. 199, 168–173.Google Scholar
Kingman, J. (1969). Random secants of a convex body. J. Appl. Prob. 6, 660–672.CrossRefGoogle Scholar
Mathai, A. (1999). An Introduction to Geometrical Probability: Distributional Aspects with Applications, vol. 1 of Statistical Distributions and Models with Applications. Gordon & Breach Science Publishers, Amsterdam.Google Scholar
Miles, R. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
Rappaport, T., Annamalai, A., Buehrer, R. and Tranter, W. (2002). Wireless communications: past events and a future perspective. IEEE Commun. Mag. 40, 148–161.CrossRefGoogle Scholar
Santaló, L. (1976). Integral Geometry and Geometric Probability. Addison-Wesley.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Schweppe, F. C. (1973). Uncertain Dynamic Systems. Prentice Hall, Englewood Cliffs, NJ.Google Scholar