Fässler, Katrin Orponen, Tuomas 2019-03-20 We show that a complete doubling metric space (X,d,μ) supports a weak 1-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points s,t∈X . This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1-Poincaré inequality. Our argument passes through the intermediate notion of a generalised pencil of curves (GPC). A GPC joining s and t is a normal 1-current T, in the sense of Ambrosio and Kirchheim, with boundary ∂T=δt−δs , support contained in a ball of radius ∼d(s,t) around {s,t} , and satisfying ∥T∥≪μ , withd∥T∥/dμ(y)≲d(s,y)/μ(B(s,d(s,y)))+d(t,y)/μ(B(t,d(t,y))).We show that the 1-Poincaré inequality implies the existence of GPCs joining any pair of points in X. Then, we deduce the existence of PCs from a recent decomposition result for normal 1-currents due to Paolini and Stepanov. application/pdf https://folia.unifr.ch/global/documents/307599 https://folia.unifr.ch/documents/307599/files/fae_mcp.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00526-019-1514-3 info:eu-repo/semantics/openAccess License undefined Calculus of Variations and Partial Differential Equations. - 2019, vol. 58, no. 2, p. 69 info:eu-repo/classification/udc/51 Metric currents and the Poincaré inequality http://purl.org/coar/resource_type/c_6501