Durand-Cartagena, EstibalitzJaramillo Aguado, Jesús ÁngelShanmugalingam, Nageswari2023-06-202023-06-202009-10https://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