We deduce explicit formulas for the intrinsic volumes of an ellipsoid in \(\mathbb {R}^d\) , \(d\ge 2\) , in terms of elliptic integrals. Namely, for an ellipsoid \({\mathcal {E}}\subset \mathbb {R}^d\) with semiaxes \(a_1,\ldots , a_d\) , we show that \( V_k({\mathcal {E}})=\kappa _k\sum \limits _{i=1}^da_i^2s_{k-1}(a_1^2,\dots ,a_{i-1}^2,a_{i+1}^2,\dots ,a_d^2) \int \limits _0^{\infty }{t^{k-1}\over (a_i^2t^2+1)\prod \limits _{j=1}^d\sqrt{a_j^2t^2+1}}\,\textrm{d}t \) for all \(k=1,\ldots ,d\) , where \(s_{k-1}\) is the \((k-1)\) th elementary symmetric polynomial and \(\kappa _k\) is the volume of the k-dimensional unit ball. Some examples of the intrinsic volumes \(V_k\) with low and high k are given where our formulas look particularly simple. As an application, we derive new formulas for the expected k-dimensional volume of random k-simplex in an ellipsoid and random Gaussian k-simplex. Bibliography: 33 titles.
