Durand-Cartagena, EstibalitzJaramillo Aguado, Jesús ÁngelShanmugalingam, Nageswari2023-06-202023-06-202009-10S. G. Bobkov, C. Houdre: Some connections between isoperimetric and Sobolev-type inequalities. (English summary) Mem. Amer. Math. Soc. 129 (616) (1997). M. Bourdon, H. Pajot Quasi-conformal geometry and hyperbolic geometry. (English summary) Rigidity in dynamics and geometry (Cambridge, 2000), 1–17, Springer, Berlin, 2002. E. Durand, J. A. Jaramillo: Pointwise Lipschitz functions on metric spaces. To appear in J. Math. Anal. Appl. (2009). G.B. Folland: Real Analysis, Modern Techniques and Their Applications. Pure and Applied Mathematics (1999). P. Hajlasz: Sobolev spaces on metric-measure spaces. Contemp. Math. 338 (2003), 173–218. P. Hajlasz, P. Koskela: Sobolev met Poincaré. (English summary) Mem. Amer. Math. Soc. 145(668) (2000). J. Heinonen, P. Koskela: A note on Lipschitz Functions, Upper Gradients, and the Poincaré Inequality. New Zealand J. Math. 28 (1999), 37–42. J. Heinonen, P. Koskela: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), 1–61. E. Järvenpää, M. Järvenpää, N. Shanmugalingam K. Rogovin, and S. Rogovin: Measurability of equivalence classes and MECp−property in metric spaces. Rev. Mat. Iberoamericana 23 (2007), 811–830. J. Kinnunen, R. Korte: Characterizations of Sobolev inequalities on metric spaces. J. Math. Anal. Appl. 344 (2008), 1093–1104. R. Korte: Geometric implications of the Poincaré inequality, Licentiate’s thesis, Helsinki University of Technology, 2006. M. Miranda: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 83 (2003), 975–1004. S. Semmes: Some Novel Types of Fractal Geometry. Oxford Science Publications (2001). S.Semmes: Finding Curves on General Spaces through Quantitative Topology, with Applications to Sobolev and Poincaré Inequalities. Selecta Math., New Series 2(2) (1996), 155–295 N. Shanmugalingam: “Newtonian Spaces: An extension of Sobolev spaces to Metric Measure Spaces” Ph. D. Thesis, University of Michigan (1999), http: math.uc.edu/~nages/papers.html. N. Shanmugalingam: Newtonian Spaces: An extension of Sobolev spaces to Metric Measure Spaces. Rev. Mat. Iberoamericana, 16 (2000), 243–279.https://hdl.handle.net/20.500.14352/44466We study a geometric characterization of ∞−Poincaré inequality. We show that a path-connected complete doubling metric measure space supports an ∞−Poincaré inequality if and only if it is thick quasi-convex. We also prove that these two equivalent properties are also equivalent to the purely analytic property that N1,∞(X) = LIP∞(X), where LIP∞(X) is the collection of bounded Lipschitz functions on X and N1,∞(X) is the Newton-Sobolev space studied in [DJ].engjournal articlehttp://www.recercat.cat/bitstream/handle/2072/47933/Pr895.pdf?sequence=1http://www.crm.cat/en/Pages/default.aspxopen access517.98Análisis funcional y teoría de operadores