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.


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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[1] Brezis, H. and Kinderlehrer, D. The smoothness of solutions to nonlinear variational inequalities,Ind. Univ. Math. J.,23, 831–844, (1974). · Zbl 0278.49011
[2] Federer, H.Geometric Measure Theory, Springer Verlag, New York, 1969. · Zbl 0176.00801
[3] Gonzalez, E., Massari, U., and Tamanini, I. Minimal boundaries enclosing a given volume,Manuscripta Math.,34, 381–395, (1981). · Zbl 0481.49035
[4] Gonzalez, E., Massari, U., and Tamanini, I. On the regularity of sets minimizing perimeter with a volume constraint,Ind. Univ. Math. J.,32, 25–37, (1983). · Zbl 0504.49026
[5] Grüter, M. Boundary regularity for solutions of a partitioning problem,Arch. Rat. Mech. Anal.,97, 261–270, (1987). · Zbl 0613.49029
[6] Giusti, E.Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, MA, 1985. · Zbl 0599.30001
[7] Laurence, P. and Stredulinsky, E.W. On quasiconvex equimeasurable rearrangement, a counterexample and an example,J. fur Reine Angew. Math.,447, 63–81, (1994). · Zbl 0848.35003
[8] Massari, U. and Miranda, M. Minimal surfaces of codimension one,Math. Studies, North Holland,91, (1984). · Zbl 0565.49030
[9] Simon, L. Lectures on geometric measure theory,Proc. Centre Math. Analysis, ANU,3, (1983). · Zbl 0546.49019
[10] Tamanini, I. Boundaries of Caccioppoli sets with Hölder-continuous normal vector,J. fur Reine Angew. Math.,334, 27–39, (1982). · Zbl 0479.49028
[11] Ziemer, W.P.Weakly Differentiable Functions, Springer-Verlag, New York, 1989, 120. · Zbl 0692.46022
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