Points with rotational ellipsoids of inertia, envelopes of hyperplanes which equally fit the system of points in Rk, and ellipsoidal billiards

Abstract

For a given system of material points in Rk which represents a sample of a full rank, we study the points for which the ellipsoid of inertia is rotational. Using an initial information about such points with a rotational ellipsoid of inertia along the line of the major axis of the central ellipsoid of inertia, we construct a pencil of confocal quadrics with the following property: all the hyperplanes for which the hyperplanar moments of inertia for the given system of points are equal, are tangent to one of the quadrics from the pencil of quadrics. We close the loop by applying the confocal pencil of quadrics to get a full description of the points with rotational hyperplanar ellipsoids of inertia. We indicate relationships of the obtained results with integrable billiards within quadrics and PCA.