In polytopes, small balls about some vertex minimize perimeter

  • Frank Morgan


In (the surface of) a convex polytope Pn in ℝn+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter.

Math Subject Classifications


Key Words and Phrases

Isoperimetric perimeter minimizing polytope 


  1. [1]
    Almgren, Jr., F. J. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Mem. Amer. Math. Soc. 4(165), (1976).Google Scholar
  2. [2]
    Bérard, P., Besson, G., and Gallot, S. Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov [An isoperimetric inequality generalizing the Paul Levy-Gromov inequality],Invent. Math. 80, 295–308, (1985).CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Cotton, A., Freeman, D., Gnepp, A., Ng, T.F., Spivack, J., and Yoder, C. The isoperimetric problem on some singular surfaces,J. Aust. Math. Soc. 78, 167–197, (2005).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Druet, O. Sharp local isoperimetric inequalities involving the scalar curvature,Proc. Amer. Math. Soc. 130, 2351–2361, (2002).CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    Duzaar, F. and Steffen, K. Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds,Indiana Univ. Math. J. 45, 1045–1093, (1996).CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    Federer, H.Geometric Measure Theory, Springer-Verlag, 1969.Google Scholar
  7. [7]
    Ghomi, M. Optimal smoothing for convex polytopes,Bull. London Math. Soc. 36, 483–492, (2004).CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Gnepp, A., Ng, T. E, and Yoder, C. Isoperimetric domains on polyhedra and singular surfaces,NSF “SMALL” Undergraduate Research Geometry Group Report, Williams College, 1998.Google Scholar
  9. [9]
    Gonzalo, J. Large soap bubbles and isoperimetric regions in the product of Euclidean space with a closed manifold, PhD thesis, Univ. Calif. Berkeley, 1991.Google Scholar
  10. [10]
    Gromov, M. Isoperimetric inequalities in Riemannian manifolds, Appendix I to Vitali D. Milman and Gideon Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces,Lect. Notes in Math. 1200, Springer-Verlag, 1986.Google Scholar
  11. [11]
    Gromov, M. Isoperimetry of waists and concentration of maps,IHES, 2002,∼gromov/topics/topic11.html.Google Scholar
  12. [12]
    Howards, H., Hutchings, M., and Morgan, F. The isoperimetric problem on surfaces,Amer. Math. Monthly 106, 430–439, (1999).CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Hutchings, M. Isoperimetric ignorance: A personal report, email to Frank Morgan, July 16, 2001.Google Scholar
  14. [14]
    Kleiner, B. cited by Per Tomter, Constant mean curvature surfaces in the Heisenberg group,Proc. Sympos. Pure Math. 54(1), 485–495, (1993).MathSciNetGoogle Scholar
  15. [15]
    Lions, P.-L. and Pacella, F. Isoperimetric inequalities for convex cones,Proc. Amer. Math. Soc. 109, 477–485, (1990).CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    Morgan, F.Geometric Measure Theory: A Beginner’s Guide, Academic Press, 3rd ed., 2000.Google Scholar
  17. [17]
    Morgan, F. Manifolds with density,Notices Amer. Math. Soc. 52, 853–858, (2005).MathSciNetMATHGoogle Scholar
  18. [18]
    Morgan, F.Riemannian Geometry: A Beginner’s Guide, A. K. Peters, 1998.Google Scholar
  19. [19]
    Morgan, F. and Johnson, D. L. Some sharp isoperimetric theorems for Riemannian manifolds,Indiana Univ. Math. J. 49, 1017–1041, (2000).CrossRefMathSciNetMATHGoogle Scholar
  20. [20]
    Morgan, F. and Ritoré, M. Isoperimetric regions in cones,Trans. Amer. Math. Soc. 354, 2327–2339, (2002).CrossRefMathSciNetMATHGoogle Scholar
  21. [21]
    Pedrosa, R.H.L. The isoperimetric problem in spherical cylinders,Ann. Global Anal. Geom. 26, 333–354, (2004).CrossRefMathSciNetMATHGoogle Scholar
  22. [22]
    Pedrosa, R. H. L. and Ritoré, M. Isoperimetric domains in the Riemannian product of a circle with a simply connected space form,Indiana Univ. Math. J. 48, 1357–1394, (1999).CrossRefMathSciNetMATHGoogle Scholar
  23. [23]
    Ros, A. The isoperimetric problem, inGlobal Theory of Minimal Surfaces (Clay Research Institution Summer School, 2001, Hoffman, D., Ed.,) Amer. Math. Soc., 2005.Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2007

Authors and Affiliations

  • Frank Morgan
    • 1
  1. 1.Department of Mathematics and StatisticsWilliams CollegeWilliamstown

Personalised recommendations