Random affine simplexes
In: Journal of Applied Probability, Jg. 56 (2019-03-01), S. 39-51
Online
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Zugriff:
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+1 μd,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).
Titel: |
Random affine simplexes
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Autor/in / Beteiligte Person: | Gusakova, Anna ; Zaporozhets, Dmitry ; Götze, Friedrich |
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Zeitschrift: | Journal of Applied Probability, Jg. 56 (2019-03-01), S. 39-51 |
Veröffentlichung: | Cambridge University Press (CUP), 2019 |
Medientyp: | unknown |
ISSN: | 1475-6072 (print) ; 0021-9002 (print) |
DOI: | 10.1017/jpr.2019.4 |
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