The integral $$\int_V \frac{d^3x\,d^3y}{|\mathbf{x}-\mathbf{y}|}$$ over various regions $V$ is of interest in physics, where it is related to the gravitational binding energy of an object with uniform density. The best-known case is for a ball. An analytic result is also known for a rectangular cuboid.
I am not aware of analytic results for platonic solids other than the cube, so I have been investing this integral numerically by Monte Carlo techniques. It appears that its value is the same to 3 digits for the dodecahedron and icosahedron, when they have the same volume, which is interesting because these are dual polyhedra. (The octahedron and cube are also dual, but I don't have results for the octahedron because of a bug in Mathematica's RandomPoints function.)
Is there any mathematical argument that this integral should be the same for dual polyhedra of equal volume?