Optimization problems in gravitational attraction

AbstractThree types of problems in gravitational attraction are considered. In the first type, two isoperimetric ones are solved elementarily by means of a known inequality concerning integral means. In the second one, it is shown that if a homogeneous ellipsoid is divided into two measurable sets, then the gravitational attractive force between them is greatest when the two sets are hemiellipsoids formed by a plane containing the two largest principal axes. In the third one, it is shown that the motion of a free particle in a straight tunnel through a homogeneous ellipsoidal planet is simple harmonic. Furthermore, the periods achieve their maximum and minimum values when the tunnels are parallel to the largest and smallest axes, respectively