On Convex Surfaces with Minimal Moment of Inertia

We investigate the problem of minimizing the moment of inertia among convex surfaces in R having a specified surface area. First we prove a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by H. Sachs and later R. R. Hall, who showed the equilateral triangle minimizes the moment of inertia, among all convex curves with given length.