On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 1

We introduce new flatness coefficients, which we call $\iota$-numbers, for Ahlfors $k$-regular sets in metric spaces ($k\in \mathbb{N}$). Using these coefficients for $k=1$, we characterize uniform $1$-rectifiability in rather general metric spaces, completing earlier work by Hahlomaa and Schul. Our proof proceeds by quantifying an isometric embedding theorem due to Menger, and by an abstract argument that allows to pass from a local covering by continua to a global covering by $1$-regular connected sets.