Existence of Isoperimetric Sets with Densities “Converging from Below” on RN

Guido De Philippis, Giovanni Franzina, Aldo Pratelli

Research output: Contribution to journalArticlepeer-review


In this paper, we consider the isoperimetric problem in the space RN with a density. Our result states that, if the density f is lower semi-continuous and converges to a limit a> 0 at infinity, with f≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331–365, 2013.

Original languageEnglish (US)
Pages (from-to)1086-1105
Number of pages20
JournalJournal of Geometric Analysis
Issue number2
StatePublished - Apr 1 2017


  • Existence of optimal sets
  • Isoperimetric problem
  • Perimeter with density

ASJC Scopus subject areas

  • Geometry and Topology


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