We deduce explicit formulae 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 \begin{ align*} V_k({ \mathcal E})=\kappa_k\sum_{ i=1}^da_i^2s_{ k-1}(a_1^2,\dots,a_{ i-1}^2,a_{ i+1}^2,\dots,a_d^2)\int_0^{ \infty}{ t^{ k-1}\over(a_i^2t^2+1)\prod_{ j=1}^d\sqrt{ a_j^2t^2+1}}\,\rm{ d}t \end{ align*} 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 formulae look particularly simple. As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.
Categories: math.MG math.PR