The Journal of Geometric Analysis

, Volume 7, Issue 4, pp 653–677 | Cite as

Area minimizing sets subject to a volume constraint in a convex set



For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center of B.

Math Subject Classifications

49Q20 49Q15 49Q10 52A20 

Key Words and Phrases

sets of finite perimeter volume constraint area minimizing constant mean curvature 


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Copyright information

© Mathematica Josephina, Inc. 1997

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Wisconsin, Center-Richland, Richland Center
  2. 2.Mathematics DepartmentIndiana UniversityBloomington

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