I have a lot of records in EM, and I know all about charge induction and Gauss's theorem for systems of conductors, nonetheless I still have a problem that I cannot face without feeling uncomfortable.
Suppose to have a hollow conducting sphere, with a pointlike charge $q$ inside, placed at a point not at the centre of the sphere. That induces an asymmetrical (but axisymmetrical) charge distribution on the inner surface of the hollow sphere; but also, a perfectly homogeneous charge distribution on the external surface. Why is this?
This is something I can understand could happen, but I miss some actual proof that it must happen. It must reside in something related to the particular symmetry of the sphere, but for me it is not enough to say that "this happens due to the spherical symmetry". Is there something that clearly forces things to happen like this?