A solution is given to a long-standing open problem posed by Almgren. Minimizing the free energy under a mass constraint may generate a convex crystal in \(n-\) dimensions subject to assumptions on the potential \(g \ge 0\) , \(g(0)=0\) . The problem attributed to Almgren is to understand if this is the case assuming g is convex or more generally if sub-level sets are convex. The initial theorem proven in the paper is that the answer is negative. Nevertheless, supposing \(n=2\) , when the potential also satisfies an integral-type assumption, the convexity is true if existence holds. In the proof, the tools include the first variation formula for the anisotropic surface energy, density estimates, geometric results in the plane related to convex hulls, and a perturbation technique. Also, a general theory for uniqueness is introduced.
