The least doubling constant of a metric measure space

The first author has been partially supported by the Spanish Government grant MTM2016- 75196-P (MINECO/FEDER, UE) and the Catalan Autonomous Government grant 2017SGR358. The second author gratefully acknowledges support of Spanish Ministerio de Economia, Industria y Competitividad through grants MTM2016-76808-P, MTM2016-75196-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). The second author wishes to thank Benoit Kloeckner (Paris-Est) and Estibalitz Durand-Cartagena (UNED) for interesting discussions on the topic of this paper.