On Volumes of Hyper-Ellipsoids

Volumes of objects in higher dimensions are of interest in higher-dimensional geometry. In this note, we give a sampling of some of the topics that are considered in this field. We first provide an alternative proof for a formula for the volumes of generalized super-ellipsoids that was obtained by Dirichlet in the 19th century. Then we employ a geometric Monte Carlo method that allows us to estimate hyper-volumes numerically. We then end the presentation with a brief discussion on the volumes of revolution in higher dimensions.