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n-spheres and n-ellipsoids

3-dim projections of 4-dim MVN

n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

In the third post of this series we have discussed an idea of I. Rivin (see [1], [2]). Rivin has shown that the surface area of a general (n-1)-dimensional ellipsoid in a n-dimensional Euclidean space can be expressed in terms of an expectation value of a specific function weighted by a multivariate Gaussian probability density [pdf]. In contrast to (n-1)-spheres… Read More »n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

In the 2nd post of this series we have derived an explicit formula for the volume of a n-dimensional ellipsoid (in an Euclidean space). One reason for the relatively simple derivation was that the determinant of the generating linear transformation could be taken in front of the volume integral. Unfortunately, an analogue sequence of steps is not possible for an… Read More »n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

Concentric surfaces of ellipsoids

n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids

This post series provides some insights into the nature of n-dimensional spheres and ellipsoids and the derivation of some special integrals over their volumes. In this 2nd post we look at the volume of a “(n-1)–ellipsoid“. This term refers to a closed ellipsoidal surface in a n-dimensional Euclidean space. Such a surface is a (n-1)-dimensional manifold. See the 1st post… Read More »n-dimensional spheres and ellipsoids – II – volume of n-ellipsoids

n-dimensional spheres and ellipsoids – I – Gamma-function, volume and surface of n-spheres

This post series summarizes some properties of spheres and ellipsoids in a n-dimensional Euclidean space. In addition we are going to study some special integrals over the volume of so called n-spheres and their surfaces. As a preparatory step we look at some useful properties of the so called Gamma-function, which almost automatically appears when one works with integrals in… Read More »n-dimensional spheres and ellipsoids – I – Gamma-function, volume and surface of n-spheres