A two-point function approach to connectedness of drops in convex potentials

We establish connectedness of volume constrained minimisers of energies involving surface tensions and convex potentials. By a previous result of McCann, this implies that minimisers are convex in dimension two. This positively answers an old question of Almgren. We also prove convexity of minimisers when the volume constraint is dropped. Our proof is based on the introduction of a new "two-point function" which measures the lack of convexity and which gives rise to a negative second variation of the energy.