of a geodesic $n$-gon $T_n$, in a geodesic metric space $X$, belongs to the closed $(1+\log_2(n-1))f_X(|T_n|)$ neighborhood of other $(n-1)$ sides, where $f_X$ is the hyperbolicity function of the geodesic metric space $X$ and $|T_n|$ is the perimeter of the $n$-gon $T_n$.

A function $f:[0,\infty)\rightarrow [0,\infty)$ is called sub-linear if $\lim_{t\rightarrow \infty }\frac{f(t)}{t}=0$.

\[hyptriangle\] A geodesic metric space $(X,d)$ is hyperbolic if and only if the hyperbolicity function $f_X$ is sub-linear.

If $X$ is $\delta$ hyperbolic space for some $\delta>0$, then by the definition of hyperbolicity function $f_X$ is bounded by $\delta$ and hence sub-linear.

Suppose the hyperbolicity function $f_X$ of $(X,d)$ is sub-linear. Lets assume that $(X,d)$ is not hyperbolic. By ${\color{blue}\ref*{asymptoticgromov}}$, there exists an asymptotic cone which is not $\mathbb{R}$-tree (Say, $Con^{\omega}(X,e,d=\{d_n\})$). Hence by definition of $\mathbb{R}$-tree, there exists a non-trivial simple loop in $Con^{\omega}(X,e,d)$. As in the definition of tree graded space ($T_1$) and ($T_2'$) are equivalent, we can assume that there exists a non-trivial simple geodesic triangle in $Con^{\omega}(X,e,d)$. According to lemma $\color{blue}{\ref*{limitgeodesictrg}}$ applied to the collection $\mathcal{P}$ of all one-element subsets of $Con^{\omega}(X,e,d)$, to show that $Con^{\omega}(X,e,d)$ is a tree it suffices

(1,0) – (5,0); (7,2) parabola (5,0); (-1,2) parabola (1,0); (7,2) parabola (4,5); (1,4) 