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The relative isoperimetric inequality outside convex domains in \(\mathbb{R}^{n}\). (English) Zbl 1116.58016

Summary: We prove that the area of a hypersurface \(\Sigma\) which traps a given volume outside a convex domain \(C\) in Euclidean space \(\mathbb{R}^{n}\) is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when \(C\) has smooth boundary \(\partial C\), we show that equality holds if and only if \(\Sigma\) is a hemisphere which meets \(\partial C\) orthogonally.

MSC:

58E35 Variational inequalities (global problems) in infinite-dimensional spaces
49Q20 Variational problems in a geometric measure-theoretic setting
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