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Isoperimetric inequalities in unbounded convex bodies
About this Title
Gian Paolo Leonardi, Manuel Ritoré and Efstratios Vernadakis
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 276, Number 1354
ISBNs: 978-1-4704-5118-9 (print); 978-1-4704-7020-3 (online)
DOI: https://doi.org/10.1090/memo/1354
Published electronically: March 1, 2022
Keywords: Isoperimetric inequalities,
isoperimetric profile,
isoperimetric regions,
convex bodies,
asymptotic cylinders,
rigidity,
isoperimetric dimension
Table of Contents
Chapters
- List of symbols
- 1. Introduction
- 2. Convex bodies and finite perimeter sets
- 3. Unbounded convex bodies of uniform geometry
- 4. A generalized existence result
- 5. Concavity of the isoperimetric profile
- 6. Sharp isoperimetric inequalities and isoperimetric rigidity
- 7. Asymptotic behavior of the isoperimetric profile of an unbounded convex body
Abstract
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb {R}^{n}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of $C$, we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.- Gabriel Acosta and Ricardo G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc. 132 (2004), no. 1, 195–202. MR 2021262, DOI 10.1090/S0002-9939-03-07004-7
- F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI 10.1090/memo/0165
- F. Alter, V. Caselles, and A. Chambolle, A characterization of convex calibrable sets in $\Bbb R^N$, Math. Ann. 332 (2005), no. 2, 329–366. MR 2178065, DOI 10.1007/s00208-004-0628-9
- Maria Athanassenas, A variational problem for constant mean curvature surfaces with free boundary, J. Reine Angew. Math. 377 (1987), 97–107. MR 887402, DOI 10.1515/crll.1987.377.97
- Hédy Attouch and Gerald Beer, On the convergence of subdifferentials of convex functions, Arch. Math. (Basel) 60 (1993), no. 4, 389–400. MR 1206324, DOI 10.1007/BF01207197
- Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084
- V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, Ph.D. thesis, Institut Fourier, 2003.
- Vincent Bayle and César Rosales, Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds, Indiana Univ. Math. J. 54 (2005), no. 5, 1371–1394. MR 2177105, DOI 10.1512/iumj.2005.54.2575
- Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513–541 (French). MR 690651
- S. G. Bobkov, On isoperimetric constants for log-concave probability distributions, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1910, Springer, Berlin, 2007, pp. 81–88. MR 2347041, DOI 10.1007/978-3-540-72053-9_{4}
- Jürgen Bokowski and Emanuel Sperner Jr., Zerlegung konvexer Körper durch minimale Trennflächen, J. Reine Angew. Math. 311(312) (1979), 80–100 (German). MR 549959, DOI 10.1515/crll.1979.311-312.80
- T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR 920366
- J. B. Bostwick and P. H. Steen, Stability of constrained capillary surfaces, Annual review of fluid mechanics. Vol. 47, Annu. Rev. Fluid Mech., vol. 47, Annual Reviews, Palo Alto, CA, 2015, pp. 539–568. MR 3727180
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 450480, DOI 10.1016/0022-1236(76)90004-5
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
- Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. Differential geometric and analytic perspectives. MR 1849187
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, NJ, 1970, pp. 195–199. MR 402831
- Jaigyoung Choe, Mohammad Ghomi, and Manuel Ritoré, Total positive curvature of hypersurfaces with convex boundary, J. Differential Geom. 72 (2006), no. 1, 129–147. MR 2215458
- Jaigyoung Choe, Mohammad Ghomi, and Manuel Ritoré, The relative isoperimetric inequality outside convex domains in $\textbf {R}^n$, Calc. Var. Partial Differential Equations 29 (2007), no. 4, 421–429. MR 2329803, DOI 10.1007/s00526-006-0027-z
- Jaigyoung Choe and Manuel Ritoré, The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds, J. Reine Angew. Math. 605 (2007), 179–191. MR 2338131, DOI 10.1515/CRELLE.2007.031
- F. H. Clarke, Optimization and nonsmooth analysis, 2nd ed., Classics in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1058436, DOI 10.1137/1.9781611971309
- Thierry Coulhon and Laurent Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), no. 2, 293–314 (French). MR 1232845, DOI 10.4171/RMI/138
- Martin Dyer and Alan Frieze, Computing the volume of convex bodies: a case where randomness provably helps, Probabilistic combinatorics and its applications (San Francisco, CA, 1991) Proc. Sympos. Appl. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1991, pp. 123–169. MR 1141926, DOI 10.1090/psapm/044/1141926
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Mouhamed Moustapha Fall, Area-minimizing regions with small volume in Riemannian manifolds with boundary, Pacific J. Math. 244 (2010), no. 2, 235–260. MR 2587431, DOI 10.2140/pjm.2010.244.235
- F. Fiala, Le problème des isopérimètres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv. 13 (1941), 293–346 (French). MR 6422, DOI 10.1007/BF01378068
- A. Figalli and E. Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal. 23 (2013), no. 2, 938–969. MR 3023863, DOI 10.1007/s12220-011-9270-4
- Robert Finn, Equilibrium capillary surfaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 284, Springer-Verlag, New York, 1986. MR 816345, DOI 10.1007/978-1-4613-8584-4
- Matteo Galli and Manuel Ritoré, Existence of isoperimetric regions in contact sub-Riemannian manifolds, J. Math. Anal. Appl. 397 (2013), no. 2, 697–714. MR 2979606, DOI 10.1016/j.jmaa.2012.08.017
- Sylvestre Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, Astérisque 163-164 (1988), 5–6, 31–91, 281 (1989) (French, with English summary). On the geometry of differentiable manifolds (Rome, 1986). MR 999971
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753, DOI 10.1512/iumj.1983.32.32003
- Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 2307192
- Michael Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rational Mech. Anal. 97 (1987), no. 3, 261–270. MR 862549, DOI 10.1007/BF00250810
- John E. Hutchinson and Yoshihiro Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory, Calc. Var. Partial Differential Equations 10 (2000), no. 1, 49–84. MR 1803974, DOI 10.1007/PL00013453
- R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), no. 3-4, 541–559. MR 1318794, DOI 10.1007/BF02574061
- E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), no. 1, 97–100 (German). MR 1512244, DOI 10.1007/BF01208645
- Ernst Kuwert, Note on the isoperimetric profile of a convex body, Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003, pp. 195–200. MR 2008339
- G. P. Leonardi and S. Rigot, Isoperimetric sets on Carnot groups, Houston J. Math. 29 (2003), no. 3, 609–637. MR 2000099
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
- Pierre-Louis Lions and Filomena Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc. 109 (1990), no. 2, 477–485. MR 1000160, DOI 10.1090/S0002-9939-1990-1000160-1
- Francesco Maggi, Sets of finite perimeter and geometric variational problems, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. MR 2976521, DOI 10.1017/CBO9781139108133
- D. Michael, Meniscus stability, Annu. Rev. Fluid Mech. 13 (1981), 189–215.
- Emanuel Milman, On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009), no. 1, 1–43. MR 2507637, DOI 10.1007/s00222-009-0175-9
- Luciano Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142. MR 866718, DOI 10.1007/BF00251230
- Andrea Mondino and Stefano Nardulli, Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions, Comm. Anal. Geom. 24 (2016), no. 1, 115–138. MR 3514556, DOI 10.4310/CAG.2016.v24.n1.a5
- Frank Morgan, Clusters minimizing area plus length of singular curves, Math. Ann. 299 (1994), no. 4, 697–714. MR 1286892, DOI 10.1007/BF01459806
- Frank Morgan, In polytopes, small balls about some vertex minimize perimeter, J. Geom. Anal. 17 (2007), no. 1, 97–106. MR 2302876, DOI 10.1007/BF02922085
- Frank Morgan, The Levy-Gromov isoperimetric inequality in convex manifolds with boundary, J. Geom. Anal. 18 (2008), no. 4, 1053–1057. MR 2438911, DOI 10.1007/s12220-008-9047-6
- Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
- Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000), no. 3, 1017–1041. MR 1803220, DOI 10.1512/iumj.2000.49.1929
- Stefano Nardulli, Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile, Asian J. Math. 18 (2014), no. 1, 1–28. MR 3215337, DOI 10.4310/AJM.2014.v18.n1.a1
- Stefano Nardulli and Pierre Pansu, A complete Riemannian manifold whose isoperimetric profile is discontinuous, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 537–549. MR 3801288, DOI 10.1007/s00028-017-0410-6
- T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules 19 (1986), no. 10, 2621–2632.
- Frank Pacard and Manuel Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom. 64 (2003), no. 3, 359–423. MR 2032110
- L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
- Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), no. 4, 1357–1394. MR 1757077, DOI 10.1512/iumj.1999.48.1614
- Ch. Pittet, The isoperimetric profile of homogeneous Riemannian manifolds, J. Differential Geom. 54 (2000), no. 2, 255–302. MR 1818180
- Manuel Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces, Math. Z. 226 (1997), no. 3, 465–481. MR 1483543, DOI 10.1007/PL00004351
- Manuel Ritoré, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere, Math. Z. 226 (1997), no. 1, 127–146. MR 1472144, DOI 10.1007/PL00004326
- Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom. 9 (2001), no. 5, 1093–1138. MR 1883725, DOI 10.4310/CAG.2001.v9.n5.a5
- Manuel Ritoré, The isoperimetric problem in complete surfaces of nonnegative curvature, J. Geom. Anal. 11 (2001), no. 3, 509–517. MR 1857855, DOI 10.1007/BF02922017
- Manuel Ritoré, Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions, Rev. Mat. Iberoam. 33 (2017), no. 1, 239–250. MR 3615450, DOI 10.4171/RMI/935
- Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv. 67 (1992), no. 2, 293–305. MR 1161286, DOI 10.1007/BF02566501
- Manuel Ritoré and Antonio Ros, The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds, Trans. Amer. Math. Soc. 348 (1996), no. 1, 391–410. MR 1322955, DOI 10.1090/S0002-9947-96-01496-1
- Manuel Ritoré and César Rosales, Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4601–4622. MR 2067135, DOI 10.1090/S0002-9947-04-03537-8
- Manuel Ritoré and Efstratios Vernadakis, Large isoperimetric regions in the product of a compact manifold with Euclidean space, Adv. Math. 306 (2017), 958–972. MR 3581323, DOI 10.1016/j.aim.2016.11.001
- Manuel Ritoré and Efstratios Vernadakis, Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 643–663. MR 3385175, DOI 10.1007/s00526-014-0800-3
- Manuel Ritoré and Efstratios Vernadakis, Isoperimetric inequalities in Euclidean convex bodies, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4983–5014. MR 3335407, DOI 10.1090/S0002-9947-2015-06197-2
- Manuel Ritoré and Efstratios Vernadakis, Isoperimetric inequalities in conically bounded convex bodies, J. Geom. Anal. 26 (2016), no. 1, 474–498. MR 3441524, DOI 10.1007/s12220-015-9559-9
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. MR 274683
- Antonio Ros, Isoperimetric inequalities in crystallography, J. Amer. Math. Soc. 17 (2004), no. 2, 373–388. MR 2051615, DOI 10.1090/S0894-0347-03-00447-8
- Antonio Ros, The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260
- Antonio Ros and Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), no. 1, 19–33. MR 1338315, DOI 10.1007/BF01263611
- César Rosales, Isoperimetric regions in rotationally symmetric convex bodies, Indiana Univ. Math. J. 52 (2003), no. 5, 1201–1214. MR 2010323, DOI 10.1512/iumj.2003.52.2320
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- Paul Steinhagen, Über die größte Kugel in einer konvexen Punktmenge, Abh. Math. Sem. Univ. Hamburg 1 (1922), no. 1, 15–26 (German). MR 3069385, DOI 10.1007/BF02940577
- Peter Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal. 101 (1988), no. 3, 209–260. MR 930124, DOI 10.1007/BF00253122
- Peter Sternberg and Kevin Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199–220. MR 1674097, DOI 10.4310/CAG.1999.v7.n1.a7
- Edward Stredulinsky and William P. Ziemer, Area minimizing sets subject to a volume constraint in a convex set, J. Geom. Anal. 7 (1997), no. 4, 653–677. MR 1669207, DOI 10.1007/BF02921639
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- E. L. Thomas, D. M. Anderson, C. S. Henkee, and D. Hoffman, Periodic area-minimizing surfaces in block copolymers, Nature 334 (1988), 598–601.
- Thomas I. Vogel, Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl. Math. 47 (1987), no. 3, 516–525. MR 889635, DOI 10.1137/0147034