Intrinsic volumes of ellipsoids

TL;DR

This paper derives explicit formulas for the intrinsic volumes of ellipsoids in any dimension using elliptic integrals, and applies these to compute expected volumes of random simplices within ellipsoids and Gaussian settings.

Contribution

It provides the first explicit elliptic integral formulas for intrinsic volumes of ellipsoids, extending geometric understanding and enabling new probabilistic calculations.

Findings

Explicit formulas for all intrinsic volumes of ellipsoids in terms of elliptic integrals.

Simplified formulas for specific low and high intrinsic volumes.

New expressions for expected volumes of random simplices in ellipsoids and Gaussian spaces.

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 $a_1,\ldots, a_d$ 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.