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Random affine simplexes

Part of: General convexity Probability theory on algebraic and topological structures Geometric probability and stochastic geometry 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]
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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 

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