Is it true that the Gaussian curvature of an ellipsoid is proportional to fourth power of the distance of the tangent plane from the center? I can verify that it holds at the places where the major axes intersect the surface. (Mathworld has an equation for the Gaussian curvature, which simplifies at those points.) But verifying that it holds elsewhere seems like it would get ugly.
The motivation for this question is that Lord Kelvin proved that the charge density on a conducting ellipsoid is proportional to the distance of the tangent plane from the center, while McAllister (I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016) finds that under certain assumptions, the charge density on a conducting surface is proportional to the fourth root of the absolute value of the Gaussian curvature. However, I think the assumptions of McAllister's result fail for the ellipsoid (actually I only have access to the abstract, so I'm not sure), so it would be nontrivial to learn that this held for the ellipsoid. (The proportionality is definitely not universal. For a pair of conducting spheres that are far apart and connected by a wire, the exponent is not 1/4. For a deep concavity, all of this definitely fails -- you get a a Faraday cage, which excludes the electric field almost completely.)