Connectedness of drops in convex potentials

An old conjecture of Almgren states that for every convex and coercive potential $g: \mathbb{R}^d\to \mathbb{R}$, every convex and one-homogeneous anisotropy $\Phi : \mathbb{R}^d\to \mathbb{R}^+$ and every volume $V>0$, the minimizers of \[ \min_{|E|=V} \int_{\partial E} \Phi(\nu) d\mathcal{H}^{d-1} + \int_{E} g dx \] are convex. I will review the known results on this problem and present recent progress obtained with G. De Philippis on the connectedness of the minimizers for smooth potentials and anisotropies. 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.