10.1017/jpr.2019.4 Götze, Friedrich Friedrich Götze Gusakova, Anna Anna Gusakova Zaporozhets, Dmitry Dmitry Zaporozhets Journal of Applied Probability Cambridge Univ Press 2019 2019-08-16T09:02:13Z 2019-09-09T13:00:19Z journal_article https://pub.uni-bielefeld.de/record/2936961 https://pub.uni-bielefeld.de/record/2936961.json 0021-9002 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).