Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.
A weighted isoperimetric inequality on the hyperbolic plane
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We prove a counterpart of the log-convex density conjecture in the hyperbolic plane.
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Free energy minimizers with radial densities: classification and quantitative stability
Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.