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Isoperimetric Inequalities in Conically Bounded Convex Bodies

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Abstract

We consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an exterior asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric profile for large volumes, and a characterization of isoperimetric regions of large volume in conically bounded convex sets of revolution are obtained.

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References

  1. Ambrozio, L.C.: Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds. J. Geom. Anal. (2013, to appear) arXiv:1301.6257

  2. Ashbaugh, M.S., Benguria, R.D.: Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature. J. London Math. Soc. (2) 52(2), 402–416 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bayle, V., Rosales, C.: Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds. Indiana Univ. Math. J. 54(5), 1371–1394 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonnesen, T., Fenchel, W.: Theory of Convex Bodies. BCS Associates, Moscow, ID (1987). Translated from the German by L. Boron, C. Christenson, B. Smith (eds.) (1987)

  5. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  6. Choe, J., Ghomi, M., Ritoré, M.: Total positive curvature of hypersurfaces with convex boundary. J. Differ. Geom. 72(1), 129–147 (2006)

    MATH  Google Scholar 

  7. Choe, J., Ghomi, M., Ritoré, M.: The relative isoperimetric inequality outside convex domains in \({ R}^n\). Calc. Var. Partial Differ. Equ. 29(4), 421–429 (2007)

    Article  MATH  Google Scholar 

  8. Choe, J., Ritoré, M.: The relative isoperimetric inequality in Cartan–Hadamard 3-manifolds. J. Reine Angew. Math. 605, 179–191 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Figalli, A., Indrei, E.: A sharp stability result for the relative isoperimetric inequality inside convex cones. J. Geom. Anal. 23(2), 938–969 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Galli, M., Ritoré, M.: Existence of isoperimetric regions in contact sub-Riemannian manifolds. J. Math. Anal. Appl. 397(2), 697–714 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gallot, S.: Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérisque, (163–164)5–6, 31–91, 281 (1989), 1988. On the geometry of differentiable manifolds (Rome, 1986)

  12. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition (2001)

  13. Giusti, E.: Minimal surfaces and functions of bounded variation. Monographs in Mathematics. Birkhäuser, Basel (1984)

    Book  Google Scholar 

  14. Gonzalez, E., Massari, U., Tamanini, I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32(1), 25–37 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Grüter, M.: Boundary regularity for solutions of a partitioning problem. Arch. Ration. Mech. Anal. 97(3), 261–270 (1987)

    Article  MATH  Google Scholar 

  16. Grüter, M., Jost, J.: Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(1), 129–169 (1986)

    MATH  MathSciNet  Google Scholar 

  17. Kuwert, E.: Note on the isoperimetric profile of a convex body. In Geometric Analysis and Nonlinear Partial Differential Equations, pp. 195–200. Springer, Berlin (2003)

  18. Leonardi, G.P., Rigot, S.: Isoperimetric sets on Carnot groups. Houston J. Math., 29(3), 609–637 (2003)

  19. Leonardi, G.P., Ritoré, M., Vernadakis, E.: Isoperimetric inequalities in unbounded convex bodies (2015, preprint)

  20. Lions, P.-L., Pacella, F.: Isoperimetric inequalities for convex cones. Proc. Am. Math. Soc. 109(2), 477–485 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Maggi, F.: Sets of finite perimeter and geometric variational problems: an introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge (2012)

  22. Ritoré, M., Rosales, C.: Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones. Trans. Am. Math. Soc., 356(11), 4601–4622 (2004)

  23. Ritoré, M., Vernadakis, E.: Isoperimetric inequalities in Euclidean convex bodies. Trans. Amer. Math. Soc. (2013, to appear). arXiv:1302.4588

  24. Ritoré, M., Vernadakis, E.: Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies (2014). arXiv:1401.3542

  25. Schneider, R.: Convex bodies: the Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  26. Sternberg, P., Zumbrun, K.: On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint. Comm. Anal. Geom. 7(1), 199–220 (1999)

    MATH  MathSciNet  Google Scholar 

  27. Ziemer, W.P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)

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Acknowledgments

Both authors have been supported by MICINN-FEDER Grant MTM2010-21206-C02-01, MINECO-FEDER grant MTM2013-48371-C2-1-P, and Junta de Andalucía Grants FQM-325 and P09-FQM-5088.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Manuel Ritoré & Efstratios Vernadakis

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  1. Manuel Ritoré
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  2. Efstratios Vernadakis
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Correspondence to Manuel Ritoré.

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Ritoré, M., Vernadakis, E. Isoperimetric Inequalities in Conically Bounded Convex Bodies. J Geom Anal 26, 474–498 (2016). https://doi.org/10.1007/s12220-015-9559-9

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  • DOI: https://doi.org/10.1007/s12220-015-9559-9

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