On the isoperimetric problem in Euclidean space with density

  • César Rosales
  • Antonio Cañete
  • Vincent Bayle
  • Frank Morgan
Original Article


We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp\((|x|^2)\) by using symmetrization techniques.


Manifolds with density Isoperimetric problem Generalized mean curvature Stability Symmetrization 

Mathematics Subject Classification (2000)

49Q20 53C17 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • César Rosales
    • 1
  • Antonio Cañete
    • 1
  • Vincent Bayle
    • 2
  • Frank Morgan
    • 3
  1. 1.Departamento de Geometría y Topología, Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Institute FourierSaint Martin D’Heres CedexFrance
  3. 3.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA

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