Journal of Applied Probability

Götze F, Gusakova A, Zaporozhets D (2019)
Journal of Applied Probability 56(1): 39-51.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
For a fixed k is an element of {1,..., d}, consider arbitrary random vectors X-0,..., X-k is an element of R-d such that the (k + 1)-tuples (UX0,..., UXk) have the same distribution for any rotation U. Let A be any nonsingular d x d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed X-i satisfies vertical bar conv (AX(0),..., AX(k))| =D (vertical bar P xi epsilon vertical bar/kk)vertical bar conv (X-0,..., X-k)|, where epsilon := {x. R-d : x(inverted perpendicular) (A (inverted perpendicular) A)(-1)x <= 1} is an ellipsoid, P xi denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace xi independent of X-0,..., X-k, and kappa(k) is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: c(k, d, p) integral(Lambda d, k) vertical bar epsilon boolean AND E vertical bar(p +d+1) mu d, k(dE) = vertical bar epsilon vertical bar(k+1) integral(Gd, k) vertical bar P-T epsilon vertical bar(p)upsilon d, k(dL), where c(k, d,p) = (k(d)(k+1) /k(k)(d+1)) (kappa(k)(d+p)+k/kappa(k)(d+p)+d). Here p>-1 and A(d, k) and G(d, 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).
Stichworte
Blaschke-Petkantschin formula; convex hull; ellipsoid; expected volume; Furstenberg-Tzkoni formula; Gaussian matrix; intrinsic volume; random; section; random simplex
Erscheinungsjahr
2019
Zeitschriftentitel
Journal of Applied Probability
Band
56
Ausgabe
1
Seite(n)
39-51
ISSN
0021-9002
eISSN
1475-6072
Page URI
https://pub.uni-bielefeld.de/record/2936961

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Götze F, Gusakova A, Zaporozhets D. Journal of Applied Probability. Journal of Applied Probability. 2019;56(1):39-51.
Götze, F., Gusakova, A., & Zaporozhets, D. (2019). Journal of Applied Probability. Journal of Applied Probability, 56(1), 39-51. doi:10.1017/jpr.2019.4
Götze, F., Gusakova, A., and Zaporozhets, D. (2019). Journal of Applied Probability. Journal of Applied Probability 56, 39-51.
Götze, F., Gusakova, A., & Zaporozhets, D., 2019. Journal of Applied Probability. Journal of Applied Probability, 56(1), p 39-51.
F. Götze, A. Gusakova, and D. Zaporozhets, “Journal of Applied Probability”, Journal of Applied Probability, vol. 56, 2019, pp. 39-51.
Götze, F., Gusakova, A., Zaporozhets, D.: Journal of Applied Probability. Journal of Applied Probability. 56, 39-51 (2019).
Götze, Friedrich, Gusakova, Anna, and Zaporozhets, Dmitry. “Journal of Applied Probability”. Journal of Applied Probability 56.1 (2019): 39-51.