Abstract: I will discuss new work with Greg Chambers on the Coulomb energy in the context of recent geometric stability results (due to Christ, Figalli, Jerison and others) for functionals that describe non-local interactions. The Coulomb energy of an electrostatic charge distribution is given by the double integral of the Newton potential against the charge density. It is known that the energy of a positive charge distribution increases under symmetrization: The physical reason is that interaction energy between the charges grows as the typical distance between them shrinks. The energy increases strictly, unless the distribution is already radially decreasing about some point. Is this characterization of equality cases "stable"? In other words, must near-maximizers be close to radially decreasing? Greg and I answer this question for charge distributions that are uniform on a set of finite positive volume. Specifically, we bound the difference of a set from a suitably translated ball in terms of the difference in Coulomb energy. Time permitting, I will sketch the proof and mention some open problems.