Abstract
In (the surface of) a convex polytope Pn in ℝn+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter.
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Almgren, Jr., F. J. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Mem. Amer. Math. Soc. 4(165), (1976).
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).
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).
Druet, O. Sharp local isoperimetric inequalities involving the scalar curvature,Proc. Amer. Math. Soc. 130, 2351–2361, (2002).
Duzaar, F. and Steffen, K. Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds,Indiana Univ. Math. J. 45, 1045–1093, (1996).
Federer, H.Geometric Measure Theory, Springer-Verlag, 1969.
Ghomi, M. Optimal smoothing for convex polytopes,Bull. London Math. Soc. 36, 483–492, (2004).
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.
Gonzalo, J. Large soap bubbles and isoperimetric regions in the product of Euclidean space with a closed manifold, PhD thesis, Univ. Calif. Berkeley, 1991.
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.
Gromov, M. Isoperimetry of waists and concentration of maps,IHES, 2002, http://www.ihes.fr/∼gromov/topics/topic11.html.
Howards, H., Hutchings, M., and Morgan, F. The isoperimetric problem on surfaces,Amer. Math. Monthly 106, 430–439, (1999).
Hutchings, M. Isoperimetric ignorance: A personal report, email to Frank Morgan, July 16, 2001.
Kleiner, B. cited by Per Tomter, Constant mean curvature surfaces in the Heisenberg group,Proc. Sympos. Pure Math. 54(1), 485–495, (1993).
Lions, P.-L. and Pacella, F. Isoperimetric inequalities for convex cones,Proc. Amer. Math. Soc. 109, 477–485, (1990).
Morgan, F.Geometric Measure Theory: A Beginner’s Guide, Academic Press, 3rd ed., 2000.
Morgan, F. Manifolds with density,Notices Amer. Math. Soc. 52, 853–858, (2005).
Morgan, F.Riemannian Geometry: A Beginner’s Guide, A. K. Peters, 1998.
Morgan, F. and Johnson, D. L. Some sharp isoperimetric theorems for Riemannian manifolds,Indiana Univ. Math. J. 49, 1017–1041, (2000).
Morgan, F. and Ritoré, M. Isoperimetric regions in cones,Trans. Amer. Math. Soc. 354, 2327–2339, (2002).
Pedrosa, R.H.L. The isoperimetric problem in spherical cylinders,Ann. Global Anal. Geom. 26, 333–354, (2004).
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).
Ros, A. The isoperimetric problem, inGlobal Theory of Minimal Surfaces (Clay Research Institution Summer School, 2001, Hoffman, D., Ed.,) Amer. Math. Soc., 2005.
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Morgan, F. In polytopes, small balls about some vertex minimize perimeter. J Geom Anal 17, 97–106 (2007). https://doi.org/10.1007/BF02922085
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DOI: https://doi.org/10.1007/BF02922085