The tangent cone to a quasimetric space with dilations

Abstract

We propound some convergence theory for quasimetric spaces that includes as a particular case the Gromov-Hausdorff theory for metric spaces. We prove the existence of the tangent cone (with respect to the introduced convergence) to a quasimetric space with dilations and, as a corollary, to a regular quasimetric Carnot-Caratheodory space. This result gives, in particular, Mitchell’s cone theorem.