A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

  • Michael Bildhauer
  • Martin Fuchs
  • Jan Müller


It is a well known fact that in \(\mathbb {R} ^n\) a subset of minimal perimeter L among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter L. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion \(G\subsetneq \mathbb {R} ^n\). Furthermore, we give a characterization of those (unbounded) convex subsets of \(\mathbb {R} ^2\) in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to \(\mathbb {R} ^n\).

Mathematics subject classification



  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  2. 2.
    Besicovitch, A.: A variant of a classical isoperimetric problem. Q. J. Math. 20(1), 84–94 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Besicovitch, A.: Variants of a classical isoperimetric problem. Q. J. Math. 3(1), 42–49 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser Boston, Boston (2006)Google Scholar
  5. 5.
    Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2013)Google Scholar
  6. 6.
    Courant, R., Hilbert, D.: Methoden der Mathematischen Physik: Erster Band. Springer, Berlin (2013)zbMATHGoogle Scholar
  7. 7.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Taylor & Francis, New York (1991)zbMATHGoogle Scholar
  8. 8.
    Fusco, N.: The stability of the isoperimetric inequality. In: Ball, J., Marcellini, P. (eds.) Vector-Valued Partial Differential Equations and Applications: Cetraro, Italy 2013, pp. 73–123. Springer, Berlin (2017)CrossRefGoogle Scholar
  9. 9.
    Giannessi, F.: Constrained Optimization and Image Space Analysis: Volume 1: Separation of Sets and Optimality Conditions. Mathematical Concepts and Methods in Science and Engineering. Springer, New York (2006)Google Scholar
  10. 10.
    Giusti, E.: The equilibrium configuration of liquid drops. J. Reine Angew. Math. 321, 53–63 (1981)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Giusti, E.: Minimal surfaces and functions of bounded variation, volume 80 of Monographs in Mathematics. Birkhäuser, Basel (1984)CrossRefGoogle Scholar
  12. 12.
    Gonzalez, E., Massari, U., Tamanini, I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32, 25–37 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lay, S.R.: Convex Sets and Their Applications. Dover books on mathematics series. Dover Publications, New York (2007)Google Scholar
  14. 14.
    Leonardi, G.P., Ritoré, M., Vernadakis, E.: Isoperimetric inequalities in unbounded convex bodies. ArXiv e-prints (2016).
  15. 15.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Morgan, F.: Geometric Measure Theory: A Beginner’s Guide. Academic Press, Cambridge (2008)zbMATHGoogle Scholar
  17. 17.
    Ritoré, M.: Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions. Rev. Mat. Iberoam. 33(1), 239–250 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Stredulinsky, E., Ziemer, W.: Area minimizing sets subject to a volume constraint in a convex set. J. Geom. Anal. 7(4), 653–677 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tamanini, I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tamanini, I.: Variational problems of least area type with constraints. Annali dell’Università’ di Ferrara 34(1), 183–217 (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tamanini, I., Giacomelli, C.: Approximation of Caccioppoli sets, with applications to problems in image segmentation. Annali dell’Università di Ferrara 35(1), 187–214 (1989)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSaarland UniversitySaarbrückenGermany

Personalised recommendations