Coarse Arcs in Metric Spaces (Musings)

Suppose $(X, d)$ is a metric space.

An arc is defined as a homeomorphism from an interval to $X$ (and its range).

Let $x, y \in X$ be any two points in an arc $J$. Then $[x, y]$ denotes the subset of the arc from $x$ to $y$.

We say $J$ is a quasi-arc if there is a number $L \geq 1$ such that $ diam ([x, y]) < L \cdot d(x, y) $ for all $x, y \in J$. This is a classical concept from geometry of metric spaces.

We propose to generalize this notion into that of coarse-arc as follows. $J$ is a $f-$ coarse arc if there exists a function $f : \mathbb{R} \to \mathbb{R} $ such that $ diam ([x, y]) < f (d(x, y)) $ for all $x, y \in J$

Moreover we say a metric space $X$ has an intrinsic curvature $f$ if any arc $J$ in $X$ can be approximated by a $f$-coarse arc and cannot be approximated by any $f’$-coarse arc such that $f’ < f$.

What is known about such coarse arcs?