Willmore‐type inequality in unbounded convex sets

Abstract In this paper, we prove the following Willmore‐type inequality: on an unbounded closed convex set ), for any embedded hypersurface with boundary satisfying a certain contact angle condition, there holds Moreover, equality holds if and only if a part of a sphere and is a part of the solid cone determined by . Here, is the bounded domain enclosed by and , is the normalized mean curvature of , and is the asymptotic volume ratio of . We also prove an anisotropic version of this Willmore‐type inequality. As a special case, we obtain a Willmore‐type inequality for anisotropic capillary hypersurfaces in a half‐space.