the minimum of ? is equal to ?1: therefore the image of X via ? is [?1, 1], and via f is, Thanks to Theorem 2.14 we know that the Lipschitz diameter is less or equal than ?, therefore we get the equality, as we wished. Bibliography [AB03] ,
Yamabe metrics on cylindrical manifolds, Geometric and Functional Analysis, vol.13, issue.2, pp.259333-198214653051, 2003. ,
DOI : 10.1007/s000390300007
The Yamabe invariants of orbifolds and cylindrical manifolds, and L[2]-harmonic spinors, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2004, issue.574, pp.121146-209911253055, 2004. ,
DOI : 10.1515/crll.2004.067
The Yamabe problem on stratied spaces, MR 3248479 [ACM15] , Hölder regularity of solutions for Schrödinger operators on stratied spaces, p.815840, 2014. ,
The signature package on Witt spaces, Annales scientifiques de l'??cole normale sup??rieure, vol.45, issue.2, 2012. ,
DOI : 10.24033/asens.2165
Équations diérentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl, vol.55, issue.9 3, pp.269296-0431287, 1976. ,
hypercontractivité et son utilisation en théorie des semigroupes , Lectures on probability theory (Saint-Flour, Lecture Notes in Math, vol.158195, pp.1114-130741347075, 1992. ,
A course in metric geometry, Graduate Studies in Mathematics, vol.33, p.183541853053, 2001. ,
DOI : 10.1090/gsm/033
Scalar curvature functions in a conformal class of metrics and conformal transformations, Transactions of the American Mathematical Society, vol.301, issue.2 ,
DOI : 10.1090/S0002-9947-1987-0882712-7
Einstein manifolds, Classics in Mathematics, p.53084, 2008. ,
Analysis and geometry of Markov diusion operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of, Mathematical Sciences], vol.348, pp.2014-3155209 ,
Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, vol.194, 1971. ,
DOI : 10.1007/bfb0064646
Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math, J, vol.85, issue.1, pp.253270-1412446, 1996. ,
Flots géométriques d'ordre quatre et pincement intégral de la courbure, 2012. ,
On the conformal scalar curvature equation and related problems , Surveys in dierential geometry Geometric ows, Surv. Differ . Geom, vol.12, pp.119-248895053077, 2008. ,
Ricci bounds for Euclidean and spherical cones, Singular phenomena and scaling in mathematical models, pp.323-3205034 ,
Yamabe ow on manifolds with edges, Math. Nachr, vol.287, issue.2-3, p.3163570, 2014. ,
Isoperimetric inequalities, Dierential geometric and analytic perspectives. MR 1849187, p.58040, 2001. ,
Spectral geometry of singular Riemannian spaces, Journal of Differential Geometry, vol.18, issue.4, p.73092058083, 1983. ,
DOI : 10.4310/jdg/1214438175
Some isoperimetric inequalities and eigenvalue estimates, Annales scientifiques de l'??cole normale sup??rieure, vol.13, issue.4, pp.419435-60828758068, 1980. ,
DOI : 10.24033/asens.1390
URL : http://www.numdam.org/article/ASENS_1980_4_13_4_419_0.pdf
Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol.9292, p.110311335035, 1990. ,
DOI : 10.1017/CBO9780511566158
Équations diérentielles caractéristiques de la sphère, Ann. Sci ,
Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, On the geometry of dierentiable manifolds, pp.163-164, 1986. ,
Riemannian geometry, 2004. ,
URL : https://hal.archives-ouvertes.fr/hal-00002870
An overview on the proof of the splitting theorem in non-smooth context, arXiv, pp.1305-4854, 2013. ,
The Yamabe constant on noncompact manifolds, J. Geom. Anal, vol.24, issue.2, p.3192307, 2014. ,
The Yamabe equation on manifolds of bounded geometry, Communications in Analysis and Geometry, vol.21, issue.5, pp.957978-3152969, 2013. ,
DOI : 10.4310/CAG.2013.v21.n5.a4
Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol.1635, p.148197046049, 1996. ,
DOI : 10.1007/BFb0092907
The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds ,
Constantes explicites pour les in??galit??s de Sobolev sur les vari??t??s riemanniennes compactes, Annales de l???institut Fourier, vol.33, issue.2, pp.151165-69949258127, 1983. ,
DOI : 10.5802/aif.921
Compact manifolds with special holonomy, p.178773353093, 2000. ,
Cones over metric measure spaces and the maximal diameter theorem, Journal de Math??matiques Pures et Appliqu??es, vol.103, issue.5, 2013. ,
DOI : 10.1016/j.matpur.2014.10.011
Scalar curvature on noncompact complete Riemannian manifolds , Nonlinear Anal, pp.1386128-97, 1996. ,
An isoperimetric comparison theorem, Inventiones Mathematicae, vol.33, issue.no. 2, pp.3747-1156385, 1992. ,
DOI : 10.1007/BF02100598
URL : http://www.digizeitschriften.de/download/PPN356556735_0108/PPN356556735_0108___log12.pdf
The Yamabe problem, Bull. Amer ,
Stratications and mappings, Dynamical systems (Proc. Sympos, pp.195232-0368064, 1971. ,
Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ, Math. J, vol.48, issue.2, pp.711748-172281453131, 1999. ,
The Local Yamabe constant of Einstein stratied spaces, pre-print available at arXiv:1411, p.7996, 2014. ,
Isoperimetric regions in cones ,
Certain conditions for a Riemannian manifold to be isometric with a sphere, Journal of the Mathematical Society of Japan, vol.14, issue.3, pp.333340-0142086, 1962. ,
DOI : 10.2969/jmsj/01430333
Isoperimetric regions in spherical cones and Yamabe constants of M × S 1 , Geom, pp.3748-2576291, 2009. ,
Analytic and geometric study of stratied spaces, Lecture Notes in Mathematics, vol.1768, p.186960158007, 2001. ,
Optimal isoperimetric inequalities for three-dimensional Cartan-Hadamard manifolds, Global theory of minimal surfaces ,
Amer, Math. Soc, vol.2, pp.395404-216726953043, 2005. ,
The isoperimetric problem, Global theory of minimal surfaces, Clay Math, Proc. Amer. Math. Soc, vol.2, pp.175-209, 2005. ,
Aspects of Sobolev-type inequalities, p.187252646048, 2002. ,
Conformal deformation of a Riemannian metric to constant
scalar curvature, Journal of Differential Geometry, vol.20, issue.2, pp.479495-78829258137, 1984. ,
DOI : 10.4310/jdg/1214439291
On the proof of the positive mass conjecture in general relativity, Communications in Mathematical Physics, vol.134, issue.1, pp.4576-52697683024, 1979. ,
DOI : 10.1007/BF01940959
Conformally flat manifolds, Kleinian groups and scalar curvature, Inventiones Mathematicae, vol.79, issue.1, pp.4771-93120458139, 1988. ,
DOI : 10.1007/BF01393992
Lecture notes prepared by, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso. MR 1333601, 1994. ,
Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata, Series 4, vol.110, issue.1, pp.353372-0463908, 1976. ,
DOI : 10.1007/BF02418013
Complete Riemannian manifolds and some vector elds, Trans. Amer. Math. Soc, vol.117, pp.251275-017402230, 1965. ,
DOI : 10.1090/s0002-9947-1965-0174022-6
Ensembles et morphismes stratiés, Bull. Amer. Math. Soc, vol.75, pp.240284-0239613, 1969. ,
Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, vol.22, issue.3, pp.265274-0240748, 1968. ,
The Aeneid: Penguin Classics Deluxe Edition , Penguin Classics Deluxe, 2006. ,
DOI : 10.4159/DLCL.virgil-aeneid.1916
Monopole metrics and the orbifold Yamabe problem ,
Sur les surfaces à courbure négative, C.R. Acad. Sci. Paris, vol.182, issue.4, 1926. ,
Complexes of manifolds, Proc. Nat. Acad. Sci. U. S. A, vol.338, pp.1011-0019306, 1947. ,