Intrinsic volumes of ellipsoids

2022/06/28 by Anna Gusakova, Gusakova, Anna, Evgeny Spodarev +3
Mathematics · #52A20 #52A38 #52A39 #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2206.14002

openalex publication_date 2022/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We deduce explicit formulae for the intrinsic volumes of an ellipsoid in \mathbb R^d, d≥ 2, in terms of elliptic integrals. Namely, for an ellipsoid \mathcal E⊂ \mathbb R^d with semiaxes a₁,…, a_d we show that V_k(\mathcal E)=κ_k∑_i=1^da_i²s_k-1(a₁²,…,a_i-1²,a_i+1²,…,a_d²)∫₀^∞t^k-1\over(a_i²t²+1)∏_j=1^d√(a_j²t²+1) \rmdt for all k=1,…,d, where s_k-1 is the (k-1)-th elementary symmetric polynomial and κ_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.

Related

• Integrals and Series Related to the Surface Area of Arbitrary Ellipsoids
• Surface area and other measures of ellipsoids
• Uniqueness of a 3D ellipsoid with given intrinsic volumes
• Simple estimates for ellipsoid measures
• Surface Area of Ellipsoids