Intrinsic volumes of ellipsoids

Abstract

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 $a_1,\dots ,a_d$ we show that \begin{align*} V_k({\mathcal E})=κ_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 $k=1,\dots ,d$, 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.