Abstract
We consider a convex solid cone mathcal {C}subset mathbb {R}^{n+1} with vertex at the origin and boundary partial mathcal {C} smooth away from 0. Our main result shows that a compact two-sided hypersurface Sigma immersed in mathcal {C} with free boundary in partial mathcal {C}setminus {0} and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape. The technique of proof also works for a non-smooth convex cone mathcal {C} provided the boundary of Sigma is away from the singular set of partial mathcal {C}.
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