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Area minimizing sets subject to a volume constraint in a convex set. (English) Zbl 0940.49025

The paper concerns the problem of minimizing the perimeter of subsets of \(\Omega\) (a convex subset of a Euclidean space) subject to a volume constraint. The problem is to determine whether in general a minimizer is convex. It has been proved that if \(\Omega\) satisfies a “great circle” condition, then any minimizer is convex.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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References:

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