article Journal of Applied Probability published yes Friedrich Götze author 10518 Anna Gusakova author 90871108 Dmitry Zaporozhets author 10020 department 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). Cambridge Univ Press2019 eng Blaschke-Petkantschin formulaconvex hullellipsoidexpected volumeFurstenberg-Tzkoni formulaGaussian matrixintrinsic volumerandomsectionrandom simplex Journal of Applied Probability 0021-9002 1475-6072 00047536560000310.1017/jpr.2019.4 56139-51 Götze, F., Gusakova, A., and Zaporozhets, D. (2019). Journal of Applied Probability. <em>Journal of Applied Probability</em> 56, 39-51. Götze, F., Gusakova, A., &amp; Zaporozhets, D., 2019. Journal of Applied Probability. <em>Journal of Applied Probability</em>, 56(1), p 39-51. Götze F, Gusakova A, Zaporozhets D (2019) <br /><em>Journal of Applied Probability</em> 56(1): 39-51. Götze, F., Gusakova, A., Zaporozhets, D.: Journal of Applied Probability. Journal of Applied Probability. 56, 39-51 (2019). F. Götze, A. Gusakova, and D. Zaporozhets, “Journal of Applied Probability”, <em>Journal of Applied Probability</em>, <strong>2019</strong>, <em>56</em>, 39-51. Götze, F., Gusakova, A., &amp; Zaporozhets, D. (2019). Journal of Applied Probability. <em>Journal of Applied Probability</em>, <em>56</em>(1), 39-51. doi:10.1017/jpr.2019.4 Götze F, Gusakova A, Zaporozhets D. Journal of Applied Probability. <em>Journal of Applied Probability</em>. 2019;56(1):39-51. <div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Götze, F., Gusakova, A., &amp; Zaporozhets, D. (2019). Journal of Applied Probability. <em>Journal of Applied Probability</em>, <em>56</em>(1), 39-51. doi:10.1017/jpr.2019.4</div> <div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Götze, F., Gusakova, A. &amp; Zaporozhets, D. (2019). Journal of Applied Probability. <em>Journal of Applied Probability</em>, <em>56</em>(1), 39-51. Cambridge Univ Press. doi:10.1017/jpr.2019.4.</div> Götze, F.; Gusakova, A.; Zaporozhets, D. (2019): Journal of Applied Probability <em>Journal of Applied Probability</em>,56:(1): 39-51. <div style="text-indent:-25px; padding-left:25px;padding-bottom:0px;">Götze, Friedrich, Gusakova, Anna, and Zaporozhets, Dmitry. 2019. “Journal of Applied Probability”. <em>Journal of Applied Probability</em> 56 (1): 39-51.</div> Götze F, Gusakova A, Zaporozhets D (2019) <br />Journal of Applied Probability.<br />Journal of Applied Probability 56(1): 39-51. Götze, Friedrich, Gusakova, Anna, and Zaporozhets, Dmitry. “Journal of Applied Probability”. <em>Journal of Applied Probability</em> 56.1 (2019): 39-51. F. Götze, A. Gusakova, and D. Zaporozhets, “Journal of Applied Probability”, <em>Journal of Applied Probability</em>, vol. 56, 2019, pp. 39-51. 29369612019-08-16T09:02:13Z2019-09-09T13:00:19Z