On the role of convexity in isoperimetry, spectral gap and concentration
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Almgren, F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. (2) 87, 321–391 (1968)
Almgren, F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc. 4(165) (1976)
Alon, N., Milman, V.D.: Concentration of measure phenomena in the discrete case and the Laplace operator of a graph. In: Israel Seminar on Geometrical Aspects of Functional Analysis (1983/84). Tel Aviv Univ., Tel Aviv (1984)
Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math., vol. 1123, pp. 177–206. Springer, Berlin (1985)
Bakry, D., Ledoux, M.: Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2), 259–281 (1996)
Barthe, F.: Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. (6) 10(3), 393–404 (2001)
Barthe, F., Kolesnikov, A.V.: Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18(4), 921–979 (2008)
Barthe, F., Maurey, B.: Some remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré Probab. Stat. 36(4), 419–434 (2000)
Barthe, F., Roberto, C.: Sobolev inequalities for probability measures on the real line. Stud. Math. 159(3), 481–497 (2003)
Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22(3), 993–1067 (2006)
Bavard, C., Pansu, P.: Sur le volume minimal de R 2. Ann. Sci. École Norm. Sup. 19(4), 479–490 (1986)
Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. Ph.D. thesis, Institut Joseph Fourier, Grenoble (2004)
Bayle, V., Rosales, C.: Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds. Indiana Univ. Math. J. 54(5), 1371–1394 (2005)
Benjamini, I., Cao, J.: A new isoperimetric comparison theorem for surfaces of variable curvature. Duke Math. J. 85(2), 359–396 (1996)
Bishop, R.L.: Infinitesimal convexity implies local convexity. Indiana Univ. Math. J. 24, 169–172 (1974/75)
Bobkov, S.: Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24(1), 35–48 (1996)
Bobkov, S.: On isoperimetric constants for log-concave probability distributions. In: Geometric Aspects of Functional Analysis, Israel Seminar 2004–2005. Lecture Notes in Math., vol. 1910, pp. 81–88. Springer, Berlin (2007)
Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)
Bobkov, S.G., Houdré, C.: Isoperimetric constants for product probability measures. Ann. Probab. 25(1), 184–205 (1997)
Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Am. Math. Soc. 129(616) (1997)
Bollobás, B., Leader, I.: Edge-isoperimetric inequalities in the grid. Combinatorica 11(4), 299–314 (1991)
Bombieri, E.: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal. 78(2), 99–130 (1982)
Brascamp, H.J., Lieb, E.H.: 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. Funct. Anal. 22(4), 366–389 (1976)
Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285. Springer, Berlin (1988)
Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15(2), 213–230 (1982)
Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton Univ. Press, Princeton (1970)
Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001)
Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Ann. Fac. Sci. Toulouse Math. (6) 15(4), 613–635 (2006)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge Univ. Press, Cambridge (1989)
Federer, H.: Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften, Band 153. Springer, New York (1969)
Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton Univ. Press, Princeton (1995)
Gallot, S.: Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérisque 163–164, 31–91 (1988). On the geometry of differentiable manifolds (Rome, 1986)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol 80. Birkhäuser, Basel (1984)
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)
Grigor’yan, A.: Isoperimetric inequalities and capacities on Riemannian manifolds. In: The Maz’ya Anniversary Collection, vol. 1, Rostock, 1998. Oper. Theory Adv. Appl., vol 109, pp. 139–153. Birkhäuser, Basel (1999)
Gromov, M.: Paul Lévy isoperimetric inequality. Preprint, I.H.E.S. (1980)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston (1999)
Gromov, M., Milman, V.D.: A topological application of the isoperimetric inequality. Am. J. Math. 105(4), 843–854 (1983)
Gromov, M., Milman, V.D.: Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compos. Math. 62(3), 263–282 (1987)
Grüter, M.: Boundary regularity for solutions of a partitioning problem. Arch. Ration. Mech. Anal. 97(3), 261–270 (1987)
Hsu, E.P.: Multiplicative functional for the heat equation on manifolds with boundary. Mich. Math. J. 50(2), 351–367 (2002)
Johnson, W.B., Schechtman, G., Zinn, J.: Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13(1), 234–253 (1985)
Kahane, J.-P.: Some Random Series of Functions, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge Univ. Press, Cambridge (1985)
Kannan, R., Lovász, L., Simonovits, M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4), 541–559 (1995)
Klartag, B.: A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Relat. Fields (to appear). arXiv:0705.0832 (2007)
Klartag, B.: Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245, 284–310 (2007)
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)
Latała, R., Wojtaszczyk, J.O.: On the infimum convolution inequality. Stud. Math. 189(2), 147–187 (2008)
Ledoux, M.: A simple analytic proof of an inequality by P. Buser. Proc. Am. Math. Soc. 121(3), 951–959 (1994)
Ledoux, M.: The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9(2), 305–366 (2000)
Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. Am. Math. Soc., Providence (2001)
Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in Differential Geometry, vol. IX, pp. 219–240. Int. Press, Somerville (2004)
Lee, M.: Isoperimetric regions in spaces. Bachelor’s degree thesis, Williams College, Williamstown, MA (2006)
Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979. Proc. Sympos. Pure Math., XXXVI, pp. 205–239. Am. Math. Soc., Providence (1980)
Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)
Maz’ja, V.G.: Classes of domains and imbedding theorems for function spaces. Dokl. Acad. Nauk SSSR 3, 527–530 (1960). Engl. transl. Sov. Math. Dokl. 1, 882–885 (1961)
Maz’ja, V.G.: p-conductivity and theorems on imbedding certain functional spaces into a C-space. Dokl. Akad. Nauk SSSR 140, 299–302 (1961). Engl. transl. Sov. Math. Dokl. 2, 1200–1203 (1961)
Maz’ja, V.G.: The negative spectrum of the higher-dimensional Schrödinger operator. Dokl. Akad. Nauk SSSR 144, 721–722 (1962). Engl. transl. Sov. Math. Dokl. 3, 808–810 (1962)
Milman, E.: On the role of convexity in functional and isoperimetric inequalities. Proc. Lond. Math. Soc. (2008). doi: 10.1112./plms/pdn045 . arxiv.org/abs/0804.0453
Milman, E.: Uniform tail-decay of Lipschitz functions implies Cheeger’s isoperimetric inequality under convexity assumptions. C. R. Math. Acad. Sci. Paris 346, 989–994 (2008)
Milman, E., Sodin, S.: An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies. J. Funct. Anal. 254(5), 1235–1268 (2008). arxiv.org/abs/math/0703857
Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math., vol. 1200. Springer, Berlin (1986). With an appendix by M. Gromov
Morgan, F.: Geometric Measure Theory (a Beginner’s Guide), 3rd edn. Academic, San Diego (2000)
Morgan, F.: Regularity of isoperimetric hypersurfaces in Riemannian manifolds. Trans. Am. Math. Soc. 355(12), 5041–5052 (2003) (electronic)
Morgan, F.: Manifolds with density. Not. Am. Math. Soc. 52(8), 853–858 (2005)
Morgan, F.: The Levy-Gromov isoperimetric inequality in convex manifolds with boundary. Manuscript, arXiv:0710.1975 (2007)
Morgan, F.: Geometric Measure Theory (a Beginner’s Guide), 4th edn. (to appear)
Morgan, F., Johnson, D.L.: Some sharp isoperimetric theorems for Riemannian manifolds. Indiana Univ. Math. J. 49(3), 1017–1041 (2000)
Paley, R.E.A.C., Zygmund, A.: A note on analytic functions in the unit circle. Proc. Camb. Philos. Soc. 28, 266–272 (1932)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Qian, Z.: A gradient estimate on a manifold with convex boundary. Proc. R. Soc. Edinb. Sect. A 127(1), 171–179 (1997)
Ros, A.: The isoperimetric problem. In: Global Theory of Minimal Surfaces. Clay Math. Proc., vol. 2, pp. 175–209. Am. Math. Soc., Providence (2005)
Schechtman, G., Zinn, J.: Concentration on the l p n ball. In: Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1745, pp. 245–256. Springer, Berlin (2000)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge Univ. Press, Cambridge (1993)
Sodin, S.: An isoperimetric inequality on the ℓ p balls. Ann. Inst. H. Poincaré Probab. Stat. 44(2), 362–373 (2008)
Sternberg, P., Zumbrun, K.: On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint. Commun. Anal. Geom. 7(1), 199–220 (1999)