2004/03/22 by Igor Rivin, Rivin, Igor
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #math.MG #math.PR
paper · pdf · doi:10.48550/arxiv.math/0403375
Supercedes preprint math.MG/0306387
arxiv created 2004/03/22 · openalex publication_date 2004/03/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this formula to derive convexity properties of the surface area, to give sharp estimates for the surface area of a large-dimensional ellipsoid, to produce asymptotic formulas for the surface area and the isoperimetric ratio of an ellipsoid in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function. We then write down general formulas for the volumes of projections of ellipsoids, and use them to extend the above-mentioned results to give explicit and approximate formulas for the higher integral mean curvatures of ellipsoids. Some of our results can be expressed as ISOPERIMETRIC results for higher mean curvatures.