[{"publisher":"Cambridge Univ Press","author":[{"last_name":"Götze","first_name":"Friedrich","id":"10518","full_name":"Götze, Friedrich"},{"first_name":"Anna","last_name":"Gusakova","full_name":"Gusakova, Anna","id":"90871108"},{"full_name":"Zaporozhets, Dmitry","first_name":"Dmitry","last_name":"Zaporozhets"}],"keyword":["Blaschke-Petkantschin formula","convex hull","ellipsoid","expected volume","Furstenberg-Tzkoni formula","Gaussian matrix","intrinsic volume","random","section","random simplex"],"external_id":{"isi":["000475365600003"]},"isi":1,"publication":"Journal of Applied Probability","abstract":[{"lang":"eng","text":"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)."}],"volume":56,"publication_identifier":{"issn":["0021-9002"],"eissn":["1475-6072"]},"article_type":"original","year":"2019","date_updated":"2019-09-09T13:00:19Z","title":"Journal of Applied Probability","quality_controlled":"1","department":[{"_id":"10020"}],"date_created":"2019-08-16T09:02:13Z","_id":"2936961","citation":{"ieee":" F. Götze, A. Gusakova, and D. Zaporozhets, “Journal of Applied Probability”, *Journal of Applied Probability*, vol. 56, 2019, pp. 39-51.","mla":"Götze, Friedrich, Gusakova, Anna, and Zaporozhets, Dmitry. “Journal of Applied Probability”. *Journal of Applied Probability* 56.1 (2019): 39-51.","apa_indent":"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

","harvard1":"Götze, F., Gusakova, A., & Zaporozhets, D., 2019. Journal of Applied Probability. *Journal of Applied Probability*, 56(1), p 39-51.","wels":"Götze, F.; Gusakova, A.; Zaporozhets, D. (2019): Journal of Applied Probability *Journal of Applied Probability*,56:(1): 39-51.","ama":"Götze F, Gusakova A, Zaporozhets D. Journal of Applied Probability. *Journal of Applied Probability*. 2019;56(1):39-51.","apa":"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","angewandte-chemie":"F. Götze, A. Gusakova, and D. Zaporozhets, “Journal of Applied Probability”, *Journal of Applied Probability*, **2019**, *56*, 39-51.","bio1":"Götze F, Gusakova A, Zaporozhets D (2019)

Journal of Applied Probability.

Journal of Applied Probability 56(1): 39-51.","frontiers":"Götze, F., Gusakova, A., and Zaporozhets, D. (2019). Journal of Applied Probability. *Journal of Applied Probability* 56, 39-51.","lncs":" Götze, F., Gusakova, A., Zaporozhets, D.: Journal of Applied Probability. Journal of Applied Probability. 56, 39-51 (2019).","default":"Götze F, Gusakova A, Zaporozhets D (2019)

*Journal of Applied Probability* 56(1): 39-51.","dgps":"Götze, F., Gusakova, A. & Zaporozhets, D. (2019). Journal of Applied Probability. *Journal of Applied Probability*, *56*(1), 39-51. Cambridge Univ Press. doi:10.1017/jpr.2019.4.

","chicago":"Götze, Friedrich, Gusakova, Anna, and Zaporozhets, Dmitry. 2019. “Journal of Applied Probability”. *Journal of Applied Probability* 56 (1): 39-51.

"},"status":"public","intvolume":" 56","issue":"1","doi":"10.1017/jpr.2019.4","user_id":"89573","page":"39-51","publication_status":"published","language":[{"iso":"eng"}],"type":"journal_article"}]