2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67695We write down estimates for the surface area, and more generally, integral mean curvatures of an ellipsoid E in n-dimensional Euclidean space in terms of the lengths of the major semi-axes. We give applications to estimating the area of parallel surfaces and volume of the tubular neighborhood of E, to the counting of lattice points contained in E and to estimating the shape of the John ellipsoid of a convex body.Metric GeometryNumber Theory52A38; 11P21text