Global minimizer for the three-dimensional angular moment of inertia

Determine the global minimizing convex surface embedded in \(\mathbb{R}^3\) for the angular moment of inertia under a fixed surface-area constraint.

Background

The paper discusses the three-dimensional analogue of the planar minimization problem studied by Sachs and Hall: minimize the angular moment of inertia, represented by the weighted perimeter with weight $|x|^2$, among convex surfaces with prescribed surface area. Previous work established existence of a minimizer and identified minimizers within selected classes, including triangular prisms, rectangular prisms, and ellipsoids. However, the unrestricted global minimization problem remains unresolved.